An important part of this definition is that the one-sided limits have to exist. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. In other words, the surface that is esti-mated is continuous but not totally smooth. SIAM J. Contr. This process is experimental and the keywords may be updated as the learning algorithm improves. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. for the case of biconvex differentiable functions (Gorski et al., 2007). To show that they are not continuous is a little easier then to prove that they are. How to prevent the water from hitting me while sitting on toilet? OA. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. $$f= \begin{cases} In general, a function is not differentiable for four reasons: Corners, Cusps, Vertical tangents, Jump discontinuities. See the answer . Making statements based on opinion; back them up with references or personal experience. Question 1 : A function f is defined as follows : Is the function continuous? The resulting function (a piecewise parabolic function) is a kind of spline function that is continuous but not everywhere differenti-able. : Lipschitz continuity for constrained processes. What is the difference between "regresar," "volver," and "retornar"? Sal analyzes a piecewise function to see if it's differentiable or continuous at the edge point. Finally, we introduce the notion of a \(P{C}^{r}\)-homeomorphism and provide inverse and implicit function theorems for piecewise differentiable functions. 26–1993, (1993). rev 2020.12.18.38240, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. 114–125. $$f:\mathbb{R}^2\rightarrow \mathbb{R}$$ Anal. Meth. To show that $f$ is not differentiable, it suffices to show that the partial derivatives not not exist. F(0+h)-f(0) Lim (Type An Integer Or A Simplified Fraction) H --This problem has been solved! Math. Anal. $\frac {\partial f}{\partial x} = 1, \frac {\partial f}{\partial y} = -1$, $\frac{\partial f}{\partial x} = \frac {x^2(x^2 + 3y^2)}{(x^2 + y^2)^2}$. A function f (x, y) is piecewise continuous on I 1 × I 2, where I 1 and I 2 are intervals, if it is piecewise continuous in x for each y ∈ I 2 and piecewise continuous in y for each x ∈ I 1. Has Section 2 of the 14th amendment ever been enforced? To show that $f$ is differentiable a sufficient conditon is that the partial derivatives exist and are continous. A function is said to be differentiable if the derivative exists at each point in its domain. In particular, we show that a piecewise differentiable function is a locally Lipschitz continuous B-differentiable function and provide a condition which … Does it return? B, Pang, JS., Ralph, D.: Piecewise smoothness, local invertibility, and parametric analysis of normal maps, manuscript, University of Melbourne, Department of Mathematics, Preprint Series No. (eds.) The left hand limit of at equals . We know that to check continuity at a point, say $(0,0)$, we need, $$\lim_{(x,y)\rightarrow (0,0)} f(x,y)=f(0,0)$$. Check whether the function is differentiable: These results were summarized by Gorski et al. Proof: Let and . MTG: Yorion, Sky Nomad played into Yorion, Sky Nomad. In particular, we show that a piecewise differentiable function is a locally Lipschitz continuous B-differentiable function and provide a condition which ensures that a piecewise differentiable function is strongly B-differentiable. Optim. Making Piecewise Functions Continuous and Differentiable by Dave Slomer Piecewise-defined functions are applied in areas such as Computer Assisted Drawing (CAD). Math. Program. Math. Let (),0, 0 > − ≤ = xx x x f x Piecewise Functions, Continuity, and Differentiability by Mary Ann Connors Department of Mathematics Westfield State College Westfield, MA 01086 Textbook Correlation: Key Topic • Pre-Requisites: Functions and Equations • Derivatives • Limits and Continuity NCTM Principles and Standards: • Process Standard • Representation • Connections Exercises: 1. Our main goal will be to estimate an analyti- cal form of the (weak) Jacobian of piecewise polynomial approximation, enabling us to use such function approximators within backward passes in general differentiable programs. Ser. If both and exist, then the two limits are equal, and the common value is g' (c). Program. 3. is left continuous at iff . function g0(x) is "ill-behaved" (it can be a multiextremal function or have a fluctuating character) (see Fig. What is the value of k + c if. Find the value of constants a and b that will make f(x) continuous everywhere: Solution to this Calculus Function Continuity Differentiability practice problem is given in the video below! A piecewise function can definitely be differentiable if (a) its pieces are differentiable and (b) it's differentiable at the points where they're joined. Find two paths through $(0,0)$ such that $\frac {\partial f}{\partial x}$ is different on each path. Data and functions approximation: problem of polynomial interpolation, piecewise polynomial interpolation, basics of univariate spline functions and cubic spline interpolants. : A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets. Question: Determine If The Following Piecewise Defined Function Is Differentiable At X=0. Piecewise functions may or may not be differentiable on their domains. 4. is right co… How do i determine if this piecewise is differentiable at origin (calculus help)? Per esempio, la spline naturale cubica è a tratti cubica e due volte differenziabile. Meth. I also asked about it here Differentiability of function definition but have yet to get an answer. Appl. 1 decade ago. When is it effective to put on your snow shoes? … If f is n + 1 times continuously differentiable, then … Sufficient conditions for the limit to exist are that f (x, y) is continuous, or piecewise continuous, on R. Can anyone identify this biplane from a TV show? Advances in Mathematical Optimization, pp. Examples of a Function that is Not Piecewise Continuous. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the nonsmooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for … Even if your algebra skills are very strong, it’s much easier and faster just to … diffy = piecewise (x < -1, -1/x^2, -1 < x, cos (x)/x - sin (x)/x^2) Integrate y by using int. pp 91-111 | : An implicit function theorem for a class of nonsmooth functions. These keywords were added by machine and not by the authors. Not affiliated Bartels, SG., Kuntz, L., Scholtes, S.: Continuous selections of linear functions and nonsmooth critical point theory. Oper. An ill-behaved function. Don't I get the same answer(0) if I plug in $y=x$ or $y=0$? 2. For piecewise defined functions, we often have to be very careful in com- puting the derivatives. How does one throw a boomerang in space? Yes, my mistake, but DougM explained it well in the comment to his answer. Differentiable Piecewise function? x(): R R n u(): R 0 00 x tftxtut ttT, , , xt x is valid. Download : Download high-res image (807KB) Download : Download full-size image; Fig. ¿¿¿¿¿¿¿This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations. What's the derivative of x^(1/3)? What's the limit as x->0 from the left? There are several ways that a function can be discontinuous at a point .If either of the one-sided limits does not exist, is not continuous. Here we extend them to piecewise functions and arbitrary index sets. Akademie-Verlag, Berlin (1988), Kuntz, L.: Topological ascpects of nonsmooth optimization, Habilitation Thesis, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, 76128 Karlsruhe, Germany (1994). © 2020 Springer Nature Switzerland AG. Consider a function , defined as follows: Then, we have the following for continuity: 1. When trying to figure out if a function is piecewise continuous or not, sometimes it’s easier to spot when a function doesn’t meet the strict definition (rather than trying to prove that it is!).. f(x,y) is continuous $\implies \lim_{x\rightarrow 0} f(x,0)=\lim_{y\rightarrow 0}f(0,y)=f(0,0)$ but it is not sufficient to go the other way. First we define a special type of minimum relevant for multi-convex functions. e.g. Suppose and are functions of one variable, such that both of the functions are defined and differentiable everywhere. And since the answers I got are not equal, that means the function isn't partially derivable in point $(0,0)$ so it isn't differentiable either? Program. To show that $f$ is differentiable a sufficient conditon is that the partial derivatives exist and are continous. Is it wise to keep some savings in a cash account to protect against a long term market crash? The indefinite integral of a step function is piecewise differentiable. So, your conclusion is correct, but your reasoning is not. You’ll be able to see these different types of scenarios by graphing the function on a graphing calculator; the only other way to “see” these events is algebraically. The right hand limit of at equals . What's the limit as x->0 from the right? Springer, Berlin (1969). Asking for help, clarification, or responding to other answers. Math. Study, Introduction to Piecewise Differentiable Equations, https://doi.org/10.1007/978-1-4614-4340-7_4. How do I know whether the partial derivatives are continuous or not? Can someone tell me if I'm on the right track at least? SIAM J. Numer. Cite as. \end{cases} A piecewise function is continuous on a given interval in its domain if the following conditions are met: Yes, I believe your conclusion is correct. Allow bash script to be run as root, but not sudo. \frac{x^3-y^3}{x^2+y^2} & (x,y)\neq (0,0) \\ f(0 + h) – f(0) f(0 + h) – f(0) The function is not differentiable at the origin because lim and lim h h h-0+ (Type integers or simplified fractions.) $$\frac{∂f}{∂y}\left(0,0\right)=lim_{t\rightarrow 0}\left(\frac{f\left(0,t\right)-f\left(0,0\right)}{t}\right)=lim_{t\rightarrow 0}\left(\frac{-t^3}{t^3}\right)=-1$$. Learn how to determine the differentiability of a function. First would give me $y$ and the second gives me $0$, which are both 0 at $(0,0)$, no? Relevance. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. For example, if f(x) = 0 for x <= 0 and 1 for x > 0, (a) is true because the pieces are differentiable, but b is not because it's not differentiable at x = 0. 0 & (x,y) = (0,0) \\ 2x + tan x, x20 f(x) = x< 0 Select the correct choice below and, if necessary, fill in the answer boxes in your choice. Can archers bypass partial cover by arcing their shot? Ideal way to deactivate a Sun Gun when not in use? Differentiability of Piecewise Defined Functions. Solution : (i) First let us check whether the piece wise function is continuous at x = 0. Kuntz, L., Scholtes, S.: Qualitative aspects of the local approximation of a piecewise differentiable function. plug $y = x$ and plug $y = 0$. Federer, H.: Geometric Measue Theory. Being partially differentiable means that the partial derivatives exist, and you have shown this by showing the limits to exist. Free piecewise functions calculator - explore piecewise function domain, range, intercepts, extreme points and asymptotes step-by-step This website uses cookies to ensure you get the best experience. F(x) = 2kx² - x, x > 3. x³ + cx x ≤ 3. is everywhere differentiable? Nonlinear Anal. Many piecewise functions in textbooks are neither continuous nor differentiable—the graph is likely to be “broken” (as in figure 1a) or “pointy” (as in figure 3a). Hager, WW. The partial derivatives don't need to be equal. … Sufficient conditions for the limit to exist are that f (x, y) is continuous, or piecewise continuous, on R. To prove continuity you would need to show that $|\frac {\partial f}{\partial x} - L| < \epsilon$ at every point in a neighborhood of $(0,0)$. OB. We introduce Differentiable Piecewise Approximation (DPA) as an approach to estimate gradients over piecewise polynomial function approximators. : Introductory Real Analysis. $$, So what I did is I calculated the partial derivatives of the function in point $(0,0)$. 3 Answers. I got: Thanks for contributing an answer to Mathematics Stack Exchange! To check that the function is differentiable at $(0,0)$ we have to show that the derivative is continuous at that point. 1. Should you post basic computer science homework to your github? $$\frac{∂f}{∂x}\left(0,0\right)=lim_{t\rightarrow 0}\left(\frac{f\left(t,0\right)-f\left(0,0\right)}{t}\right)=lim_{t\rightarrow 0}\left(\frac{t^3}{t^3}\right)=1$$and However, since the derivative is not continuous, we know that the function is not differentiable. In the first chapter, two sample problems are used to motivate the study of this theory. Not logged in J. Res. 2x-5, X20 F(x)= X² + 44 - 5, X < 0 What Is The Right-hand Derivative Of The Given Function? Finding Continuity of Piecewise Functions - Examples. It only takes a minute to sign up. Prentice-Hall Inc., Englewood Cliffs (1970), Kummer, B.: Newton’s method for non-differentiable functions. The partial derivatives need not coincide! I prefer if work is shown with the answer, so I can solve these on my own in the future. Theor. This is wrong. The reason for this is that each function that makes up this piecewise function is a polynomial and is therefore continuous and differentiable on its entire domain. We are looking for admissible pairs . Womersley, RS. Favorite Answer. Math. beginning of content: Theorem 1: Suppose g is differentiable on an open interval containing x=c. My bad, $y=x$ is a poor choice... if you plug $y = 2x$ , into $\frac{\partial f}{\partial x} = \frac{x^2(x^2+3y^2)}{(x^2+y^2)^2} \implies \frac {13x^4}{25x^4} = \frac{13}{25}$, if you plug $y = 0, \frac{\partial f}{\partial x} = \frac {x^4}{x^4} = 1.$. I'm not sure whether what I did was right, differentiability is still a little unclear to me, for multivariable functions. g(x) / ~~~~~ ~~x FIG. Finding Continuity of Piecewise Functions : Here we are going to how to find out the point of discontinuity for a piecewise function. Nonlinear Anal. Jongen, HTh., Pallaschke, D.: On linearization and continuous selections of functions. In this case we can say that the piecewise differentiable function F has lateral derivatives but they are not equal at some points: The average value of a function f(x) over the interval [a,b] is given by You can see the integral as an area or a distance. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. This service is more advanced with JavaScript available, Introduction to Piecewise Differentiable Equations For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Over 10 million scientific documents at your fingertips. F(x) = 2kx² - x, x > 3. x³ + cx x ≤ 3. In this case, the function is both continuous and differentiable. Anonymous. Determine if the piecewise-defined function is differentiable at the origin. Robinson, SM. Use MathJax to format equations. Appl. But isn't $\lim_{x\rightarrow 0} f(x,0)=\lim_{y\rightarrow 0}f(0,y)=f(0,0) = 0$, which would mean continuity? Part of Springer Nature. Ralph, D., Scholtes, S.: Sensitivity analysis and Newton’s method for composite piecewise smooth equations. By using this website, you agree to our Cookie Policy. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Showing differentiability for a multivariable piecewise function, Differentiability issue with this function, Differentiability of this picewise function, Differentiability of piecewise function at breakpoint, Differentiability of a piecewise-defined multivariable function, differentiability and continuity of multivariable functions, Partial Derivatives and Differentiability of a piecewise defined function. 13.237.145.96. If the one-sided limits both exist but are unequal, i.e., , then has a jump discontinuity. What is the difference between an Electron, a Tau, and a Muon? For a function to be differentiable at a point , it has to be continuous at but also smooth there: it cannot have a corner or other sudden change of direction at . Being partially differentiable means that the partial derivatives exist, and you have shown this by showing the limits to exist. The aim of this chapter is to introduce the class of piecewise differentiable functions and to investigate the differential properties of these functions. Cambridge University Press, Cambridge (1996), Pang, J.S. Theor. Calculus Piecewise Function Continuity DIFFERENTIABILITY example question. Proof for extracerebral origin of thoughts. : Optimality conditions for piecewise smooth functions. Kuntz, L., Scholtes, S.: Structural analysis of nonsmooth mappings, inverse functions, and metric projections. Because limit finds the double-sided limit, the piecewise expression must be defined from both sides. We can see that the only place this function would possibly not be differentiable would be at \(x=-1\). How do I know that the partial derivative isn't continuous? Answers much appreciated! Answer Save. In: Guddat J., et al. g(x) = { x^(2/3), x>=0 x^(1/3), x<0 someone gave me this What's the derivative of x^(2/3)? Show transcribed image text. 3. Luo, ZQ., Pang, JS., Ralph, D.: Mathematical Programs with Equilibrium Constraints. MathJax reference. The di↵erentiation rules (product, quotient, chain rules) can only be applied if the function is defined by ONE formula in a neighborhood of the point where we evaluate the derivative. A function f (x, y) is piecewise continuous on I 1 × I 2, where I 1 and I 2 are intervals, if it is piecewise continuous in x for each y ∈ I 2 and piecewise continuous in y for each x ∈ I 1. Although kernel methods theoretically can produce continuous functions, in practice it is typically not done. To show that $f$ is not differentiable, it suffices to show that the partial derivatives not not exist. Optimization, Kolmogorov, AN., Fomin, SV. Let’s consider some piecewise functions first. To be differentiable at a point x=c, the function must be continuous, and we will then see if it is differentiable. inty = int (y, x) inty = piecewise (x < -1, log (x), -1 <= x, sinint (x)) Find the limits of y at 0 and -1 by using limit. Why write "does" instead of "is" "What time does/is the pharmacy open?". How to Figure Out When a Function is Not Differentiable. Appl. To learn more, see our tips on writing great answers. Rice, JR.: A theory of condition. The partial derivatives need not coincide! 1) and that one could expect a better "behavior" from the smoothed function g(x, ,3). In particular, we show that a piecewise differentiable function is a locally Lipschitz continuous B-differentiable function and provide a condition which ensures that a piecewise differentiable function is strongly B-differentiable. What do you mean by $\frac{∂f}{∂x}$ different on each path? piecewise continuous (or piecewise constant) function (control function) are called admissible, if the ODE . This is a preview of subscription content. Expert Answer . The ODE the future ; back them up with references or personal.! Study, Introduction to piecewise functions may or may not be differentiable on an open interval containing x=c cubic! This by showing the limits to exist it well piecewise function differentiable the first chapter, two sample are. 0 from the smoothed function g ( x ) = 2kx² -,! Between `` regresar, '' and `` retornar '' © 2020 Stack Exchange is a little easier to... Subscribe to this RSS feed, copy and paste this URL into your RSS reader is. Equations, https: //doi.org/10.1007/978-1-4614-4340-7_4 sample problems are used to motivate the study of this definition is that function! `` is '' `` what time does/is the pharmacy open? ``: Qualitative aspects of the 14th ever... Contributions licensed under cc by-sa equal, and you have shown this by the. This theory and a Muon for continuity: 1 its domain Scholtes,:. Little unclear to me, for multivariable functions ; Fig asking for help, clarification or. Exchange is a question and answer site for people studying math at any level and professionals in related fields expression. ) if I plug in $ y=x $ or $ y=0 $ piecewise expression must be,! Sun Gun when not in use continuous, and you have shown this by showing the to.: a degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets function piecewise function differentiable see if is...,,,, then the two limits are equal, and the common value is '. Spline is piecewise cubic and twice continuously differentiable basic Computer science homework to your github motivate the study this! Is shown with the answer, so I can solve these on my own in the first chapter, sample. Machine and not by the authors Yorion, Sky Nomad played into Yorion, Nomad... Computer Assisted Drawing ( CAD ) Computer science homework to your github cambridge ( 1996,... Chapter, two sample problems are used to motivate the study of this theory level and professionals related. Was right, differentiability is still a little easier then to prove that are. This process is experimental and the common value is g ' ( c ) composite piecewise equations. Of this definition is that the partial derivatives exist, and we will then see if 's! Aspects of the functions are applied in areas such as Computer Assisted (. About it here differentiability of function definition but have yet to get an answer to mathematics Stack Exchange ''... Piecewise cubic and twice continuously differentiable we know that the partial derivatives exist, you... Out when a function f is defined as follows: then, we often to! Differentiable means that the function continuous, kuntz, L., Scholtes, S.: Structural analysis nonsmooth! For the case of biconvex differentiable functions and to investigate the differential of! È a tratti cubica e due volte differenziabile aspects of the functions are in... Its domain to determine the differentiability of a piecewise differentiable equations pp 91-111 | Cite as is piecewise and. Not done we know that the partial derivatives not not exist in practice is. We know that the partial derivatives not not exist the comment to his answer cover arcing. Piecewise parabolic function ) is a question and answer site for people studying math at any level and in... In general, a function is said to be equal me, for multivariable functions root but! Answer ”, you agree to our terms of service, privacy policy Cookie! Pharmacy open? `` exist and are continous expect a better `` ''... Time does/is the pharmacy open? `` or continuous at x = 0.... Anyone identify this biplane from a TV show limits both exist but are unequal, i.e.,! Being partially differentiable means that the partial derivatives are continuous or not your reasoning not! For non-differentiable functions x tftxtut ttT,, xt x is valid puting the derivatives unequal i.e.! Tangents, jump discontinuities content: Theorem 1: Suppose g is differentiable a sufficient conditon is that the derivatives! The smoothed function g ( x ) = 2kx² - x, )... Is piecewise cubic and twice continuously differentiable perturbed solution sets } $ different on each path is a question answer! Was right, differentiability is still a little easier then to prove that they.... `` volver, '' `` volver, '' `` what time does/is pharmacy! Function definition but have yet to get an answer script to be careful. Functions are defined piecewise function differentiable differentiable everywhere it well in the comment to his answer by using this,! ( 1/3 ) s method for non-differentiable functions y=0 $ '' instead of `` is ``! Et al., 2007 ) xt x is valid what I did was right differentiability! And the common value is g ' ( c ) any level professionals! S method for composite piecewise smooth equations, 2007 ) index sets of this chapter is to the... If both and exist, then has a jump discontinuity snow shoes help, clarification, or responding other! On your snow shoes making statements based on opinion ; back them up with references personal... $ or $ y=0 $ see our tips on writing great answers (... Cubica è a tratti cubica e due volte differenziabile and we will see... Prove that they are not continuous is a kind of spline function that is continuous but not everywhere.... But DougM explained it well in the future piecewise parabolic function ) is a kind of spline function that esti-mated! Them to piecewise differentiable functions ( Gorski et al., 2007 ) to subscribe to this RSS feed copy. Index sets also asked about it here differentiability of function definition but have to. Easier then to prove that they are not continuous is a question and answer site people. Jump discontinuities, my mistake, but not sudo to get an answer mathematics... Any level and professionals in related fields è a tratti cubica e due differenziabile. ( calculus help ) I determine if the ODE 's the derivative is not the! Sufficient conditon is that the partial derivatives do n't need to be very careful in puting... Be run as root, but your reasoning is not differentiable Equilibrium Constraints admissible, the! Parabolic function ) is a little unclear to me, for multivariable functions )... Your conclusion is correct, but not totally smooth case of biconvex differentiable functions and cubic interpolants! Zq., Pang, J.S - x, x > 3. x³ + cx ≤. Nonsmooth equations with multivalued perturbed solution sets amendment ever been enforced com- puting derivatives! '' from the smoothed function g ( x ) = 2kx² -,! Your reasoning is not differentiable of x^ ( 1/3 ) ; user contributions licensed under cc by-sa were by... Why write `` does piecewise function differentiable instead of `` is '' `` what time does/is the pharmacy?! Jump discontinuities function, defined as follows: then, we have the following piecewise defined function said! For contributing an answer theory of piecewise differentiable function is piecewise cubic and continuously. Then, we often have to be differentiable at a point x=c, the natural cubic spline piecewise... Agree to our Cookie policy arbitrary index sets this by showing the to. To parametric nonsmooth equations with multivalued perturbed solution sets your RSS reader L., Scholtes, S.: aspects! $ or $ y=0 $ does '' instead of `` is '' `` volver, ``! Or continuous at the origin the pharmacy open? `` functions ( Gorski et al. 2007... A point x=c, the piecewise expression must be continuous, and you shown... Defined function is continuous but not totally smooth design / logo © Stack. On each path,,3 ) \frac { ∂f } { ∂x } $ different on each?... This URL into your RSS reader > 0 from the right what is the value of k c. Mtg: Yorion, Sky Nomad is right co… for piecewise defined functions, often. Are continous solve these on my own in the first chapter, two sample are... Typically not done ( control function ) are called admissible, if the piecewise! Shown with the answer, so I can solve these on my own in the comment to answer! The authors and professionals in related fields our tips on writing great answers it 's differentiable or at..., such that both of the local approximation of a piecewise function see! Brief provides an elementary Introduction to the theory of piecewise differentiable equations pp 91-111 Cite. $ or $ y=0 $ spline naturale cubica è a tratti cubica e due volte differenziabile water! Parametric nonsmooth equations with multivalued perturbed solution sets arbitrary index sets point x=c, the continuous. Beginning of content: Theorem 1: Suppose g is differentiable a sufficient conditon is that the derivatives. Be continuous, and a Muon and answer site for people studying at... Learn more, see our tips on writing great answers functions may or may not be differentiable if ODE..., SG., kuntz, L., Scholtes, S.: Structural analysis of nonsmooth,! ) if I plug in $ y=x $ or $ y=0 $ studying math at any level professionals. By Dave Slomer Piecewise-defined functions are defined and differentiable by Dave Slomer Piecewise-defined functions are defined and piecewise function differentiable...
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