# how to prove a function is not differentiable

The aim of this thesis is to study the following three problems: 1) We are concerned with the behavior of normal cones and subdifferentials with respect to two types of convergence of sets and functions: Mosco and Attouch-Wets convergences. If you take the limit from the left and right (which is #1), it must equal the value of f(x) at c (which is #2). Say, if the function is convex, we may touch its graph by a Euclidean disc (lying in the épigraphe), and in the point of touch there exists a derivative. To be differentiable at a certain point, the function must first of all be defined there! By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. We introduce shrinkage estimators with differentiable shrinking functions under weak algebraic assumptions. So f is not differentiable at x = 0. 2. Why is L the derivative of L? Therefore, the function is not differentiable at x = 0. Neither continuous not differentiable. If F not continuous at X equals C, then F is not differentiable, differentiable at X is equal to C. So let me give a few examples of a non-continuous function and then think about would we be able to find this limit. How does one throw a boomerang in space? Why write "does" instead of "is" "What time does/is the pharmacy open?". The graph has a vertical line at the point. So to prove that a function is not differentiable, you simply prove that the function is not continuous. They've defined it piece-wise, and we have some choices. This fact, which eventually belongs to Lebesgue, is usually proved with some measure theory (and we prove that the function is differentiable a.e.). This fact is left without proof, but I think it might be useful for the question. Moreover, example 3, page 74 of Do Carmo's says : Let $S_1$ and $S_2$ be regular surfaces. It's saying, if you pick any x value, if you take the limit from the left and the right. Please Subscribe here, thank you!!! NOTE: Although functions f, g and k (whose graphs are shown above) are continuous everywhere, they are not differentiable at x = 0. if and only if f' (x 0 -) = f' (x 0 +). My attempt: Since any linear map on $R^3$ can be represented by a linear transformation matrix , it must be differentiable. Now, both $x$ and $L$ are differentiable , however , $x^{-1}$ is not necessarily differentiable. Join Yahoo Answers and get 100 points today. Assume that $S_1\subset V \subset R^3$ where $V$ is an open subset of $R^3$, and that $\phi:V \rightarrow R^3$ is a differentiable map such that $\phi(S_1)\subset S_2$. From the Fig. If f is differentiable at a point x 0, then f must also be continuous at x 0.In particular, any differentiable function must be continuous at every point in its domain. 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. How can you make a tangent line here? 1. I have a very vague understanding about the very step needed to show $dL=L$. Secondly, at each connection you need to look at the gradient on the left and the gradient on the right. You can't find the derivative at the end-points of any of the jumps, even though the function is defined there. Example 1: H(x)= 0 x<0 1 x ≥ 0 H is not continuous at 0, so it is not diﬀerentiable at 0. 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. Since every differentiable function is a continuous function, we obtain (a) f is continuous on [−5, 5]. Rolle's Theorem. Since $f$ is discontinuous for $x neq 0$ it cannot be differentiable for $x neq 0$. A function is said to be differentiable if the derivative exists at each point in its domain. Learn how to determine the differentiability of a function. In this video I prove that a function is differentiable everywhere in the complex plane, in other words, it is entire. Can one reuse positive referee reports if paper ends up being rejected? Firstly, the separate pieces must be joined. Differentiable functions defined on a regular surface, A differentiable map doesn't depend on the parametrization, Prove that orientable surface has differentiable normal vector, Differential geometry: restriction of differentiable map to regular surface is differentiable. A function having directional derivatives along all directions which is not differentiable. Transcript. MathJax reference. - [Voiceover] Is the function given below continuous slash differentiable at x equals three? Did the actors in All Creatures Great and Small actually have their hands in the animals? Is there a significantly different approach? The given function, say f(x) = x^2.sin(1/x) is not defined at x= 0 because as x → 0, the values of sin(1/x) changes very 2 fast , this way , sin(1/x) though bounded but not have a definite value near 0. Can you please clarify a bit more on how do you conclude that L is nothing else but the derivative of L ? exists if and only if both. When is it effective to put on your snow shoes? Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable. If any one of the condition fails then f' (x) is not differentiable at x 0. If you take the limit from the left and right (which is #1), it must equal the value of f(x) at c (which is #2). At x=0 the function is not defined so it makes no sense to ask if they are differentiable there. Plugging in any x value should give you an output. Contrapositive of the above theorem: If function f is not continuous at x = a, then it is not differentiable at x = a. Use MathJax to format equations. But when you have f(x) with no module nor different behaviour at different intervals, I don't know how prove the function is differentiable … Why are 1/2 (split) turkeys not available? Is it permitted to prohibit a certain individual from using software that's under the AGPL license? Thanks in advance. For example, the graph of f (x) = |x – 1| has a corner at x = 1, and is therefore not differentiable at that point: Step 2: Look for a cusp in the graph. Step 1: Check to see if the function has a distinct corner. Has Section 2 of the 14th amendment ever been enforced? By definition I have to show that for any local parametrization of S say $(U,x)$, map defined by $x^{-1}\circ L \circ x:U\rightarrow U$ is differentiable locally. Using three real numbers, explain why the equation y^2=x ,where x is a non   - negative real number,is not a function.. If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. Not $C^1$: Notice that $D_1 f$ does not exist at $(0,y)$ for any $y\ne 0$. "Because of its negative impacts" or "impact", Trouble with the numerical evaluation of a series, Proof for extracerebral origin of thoughts, Identify location (and painter) of old painting. It is the combination (sum, product, concettation) of smooth functions. How can I convince my 14 year old son that Algebra is important to learn? Thanks for contributing an answer to Mathematics Stack Exchange! Ex 5.2, 10 (Introduction) Greatest Integer Function f(x) = [x] than or equal to x. Making statements based on opinion; back them up with references or personal experience. Hi @Bebop. First of all, if $x:U\subset \mathbb R^2\rightarrow S$ is a parametrization, then $x^{-1}: x(U) \rightarrow \mathbb R^2$ is differentiable: indeed, following the very definition of a differentiable map from a surface, $x$ is a parametrization of the open set $x(U)$ and since $x^{-1}\circ x$ is the identity map, it is differentiable. 1. 3. Same thing goes for functions described within different intervals, like "f(x)=x 2 for x<5 and f(x)=x for x>=5", you can easily prove it's not continuous. Let me explain how it could look like. We also prove that the Kadec-Klee property is not required when the Chebyshev set is represented by a finite union of closed convex sets. It only takes a minute to sign up. We prove that $$h$$ defined by $h(x,y)=\begin{cases}\frac{x^2 y}{x^6+y^2} & \text{ if } (x,y) \ne (0,0)\\ 0 & \text{ if }(x,y) = (0,0)\end{cases}$ has directional derivatives along all directions at the origin, but is not differentiable … f(x)=[x] is not continuous at x = 1, so it’s not differentiable at x = 1 (there’s a theorem about this). Both continuous and differentiable. Differentiable, not continuous. Still have questions? Click hereto get an answer to your question ️ Prove that if the function is differentiable at a point c, then it is also continuous at that point Rolle's Theorem states that if a function g is differentiable on (a, b), continuous [a, b], and g (a) = g (b), then there is at least one number c in (a, b) such that g' (c) = 0. but i know u can tell if its a function by the virtical line test, if u graph it and u draw a virtical line down at any point and it hits the line more then once its not a function, or if u only have points then if the domain(x) repeats then its not a function. To learn more, see our tips on writing great answers. Why is a 2/3 vote required for the Dec 28, 2020 attempt to increase the stimulus checks to $2000? Therefore, by the Mean Value Theorem, there exists c ∈ (−5, 5) such that. It is given that f : [-5,5] → R is a differentiable function. (How to check for continuity of a function).Step 2: Figure out if the function is differentiable. Then the restriction$\phi|S_1: S_1\rightarrow S_2$is a differentiable map. If it isn’t differentiable, you can’t use Rolle’s theorem. Common mistakes to avoid: If f is continuous at x = a, then f is differentiable at x = a. The function is not continuous at the point. What months following each other have the same number of days? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Continuous, not differentiable. It should approach the same number. How to Check for When a Function is Not Differentiable. From the above statements, we come to know that if f' (x 0 -) ≠ f' (x 0 +), then we may decide that the function is not differentiable at x 0. If any one of the condition fails then f' (x) is not differentiable at x 0. Cruz reportedly got$35M for donors in last relief bill, Cardi B threatens 'Peppa Pig' for giving 2-year-old silly idea, These 20 states are raising their minimum wage, 'Many unanswered questions' about rare COVID symptoms, ESPN analyst calls out 'young African American' players, Visionary fashion designer Pierre Cardin dies at 98, Judge blocks voter purge in 2 Georgia counties, More than 180K ceiling fans recalled after blades fly off, Bombing suspect's neighbor shares details of last chat, 'Super gonorrhea' may increase in wake of COVID-19, Lawyer: Soldier charged in triple murder may have PTSD. Plugging in any x value should give you an output. if and only if f' (x 0 -) = f' (x 0 +) . So the first is where you have a discontinuity. $(3)\;$ The product of two differentiable functions on $\mathbb{R}^n$ is differentiable on $\mathbb{R}^n$. exist and f' (x 0 -) = f' (x 0 +) Hence. This is again an excercise from Do Carmo's book. If the function is ‘fine’ except some critical points calculate the differential quotient there Prove that it is complex differentiable using Cauchy-Riemann The function is defined through a differential equation, in a way so that the derivative is necessarily smooth. Can archers bypass partial cover by arcing their shot? A function is only differentiable only if the function is continuous. The function is differentiable from the left and right. which means that you send a vector of $\mathbb R^2$ onto $T_pS$ using the parametrization $x$ (it always gives you a good basis of the tangent space), then L acts and you read the information again using the second parametrization $y$ that takes the new vector onto $\mathbb R^2$. How to arrange columns in a table appropriately? To see this, consider the everywhere differentiable and everywhere continuous function g (x) = (x-3)* (x+2)* (x^2+4). As in the case of the existence of limits of a function at x 0, it follows that. MTG: Yorion, Sky Nomad played into Yorion, Sky Nomad. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Click hereto get an answer to your question ️ Prove that the greatest integer function defined by f(x) = [x],0 c+ and x-> c- exists. The graph has a sharp corner at the point. Can anyone give me some help ? So this function is not differentiable, just like the absolute value function in our example. This function f(x) = x 2 – 5x + 4 is a polynomial function.Polynomials are continuous for all values of x. What does 'levitical' mean in this context? 3. I do this using the Cauchy-Riemann equations. Other problem children. The derivative is defined by $f’(x) = \lim h \to 0 \; \frac{f(x+h) - f(x)}{h}$ To show a function is differentiable, this limit should exist. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the restriction $L|S$ is a differentiable map and $$dL_p(w)=L(w), p\in S,w\in T_p(S).$$. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. (Tangent Plane) Do Carmo Differential Geometry of Curves and Surfaces Ch.2.4 Prop.2. $L(p)=y(0)$. So $L$ is nothing else but the derivative of $L:S\rightarrow S$ as a map between two surfaces. exist and f' (x 0 -) = f' (x 0 +) Hence. 1. How to Prove a Piecewise Function is Differentiable - Advanced Calculus Proof Asking for help, clarification, or responding to other answers. A cusp is slightly different from a corner. ? 3. 10.19, further we conclude that the tangent line is vertical at x = 0. The converse does not hold: a continuous function need not be differentiable.For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly. Now, let $p$ be a point on the surface $S$, $x:U\subset \mathbb R^2\rightarrow S$ be a parametrization s.t. It is also given that f'( x) does not … $x(0)=p$ and $y:V\subset \mathbb R^2\rightarrow S$ be another parametrization s.t. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. That means the function must be continuous. Greatest Integer Function [x] Going by same Concept Ex 5.2, 10 Prove that the greatest integer function defined by f (x) = [x], 0 < x < 3 is not differentiable at =1 and = 2. Step 1: Find out if the function is continuous. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Get your answers by asking now. tells us there is no possibility for a tangent line there. $(4)\;$ The sum of two differentiable functions on $\mathbb{R}^n$ is differentiable on $\mathbb{R}^n$. Roughly speaking, this map does : $$\mathbb R^2 \underset{dx}{\longrightarrow} T_pS \underset{L}{\longrightarrow} T_{L(p)}S\underset{dy^{-1}}{\longrightarrow} \mathbb R^2$$ You can only use Rolle’s theorem for continuous functions. Moreover, you can easily check using the chain rule that $$df_0=d(y^{-1})_{L(p)}\circ L \circ dx_0.$$ Does it return? Understanding dependent/independent variables in physics. As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is "heading towards". Allow bash script to be run as root, but not sudo. which is clearly differentiable. In fact, this has to be expected because you might know that the derivative of a linear map between two vector spaces does not depend on the point and is equal to itself, so it has to be the same for surface or submanifold in general. Here are some more reasons why functions might not be differentiable: Step functions are not differentiable. If a function is differentiable, it is continuous. Now one of these we can knock out right from the get go. Can anyone help identify this mystery integrated circuit? https://goo.gl/JQ8Nys How to Prove a Function is Complex Differentiable Everywhere. How critical to declare manufacturer part number for a component within BOM? Figure $$\PageIndex{6}$$: A function $$f$$ that is continuous at $$a= 1$$ but not differentiable at $$a = 1$$; at right, we zoom in on the point $$(1, 1)$$ in a magnified version of the box in the left-hand plot. How to convert specific text from a list into uppercase? Is this house-rule that has each monster/NPC roll initiative separately (even when there are multiple creatures of the same kind) game-breaking? Your prove for differentiability is okay. (b) f is differentiable on (−5, 5). So $f(u,v)=y^{-1}\circ L \circ x(u,v)$ looks like $$f(u,v)=y^{-1}\circ L \circ x(u,v)=\\\ \begin{pmatrix}\varphi_1(ax_1(u,v)+bx_2(u,v)+cx_3(u,v),\cdots,gx_1(u,v)+hx_2(u,v)+ix_3(u,v)) \\ \varphi_2(gx_1(u,v)+hx_2(u,v)+ix_3(u,v),\cdots,gx_1(u,v)+hx_2(u,v)+ix_3(u,v))\end{pmatrix}$$ @user71346 Use the definition of differentiation. Limits of a function is defined there an excercise from Do Carmo 's says: Let S_1! Amendment ever been enforced a very vague understanding about the very step needed to show dL=L! Finite union of closed convex sets Stack Exchange is a differentiable map $R^3\rightarrow$... Prove a function $( 2 ) \ ;$ Every constant is! Separately ( even when there are multiple Creatures of the 14th amendment been! T differentiable, it is not necessary that the function is defined there derivative at the end-points of any the! Find out if the function is differentiable s theorem for continuous functions Voiceover ] is the function is.... Can I convince my 14 year old son that Algebra is important to learn –... ) =y ( 0 ) =p $and$ y: V\subset \mathbb s... X=0 the function given below continuous slash differentiable at x 0 - ) = f ' ( ). Is important to learn more, see our tips on writing great answers for people studying math at level. Any of the same number of days year old son that Algebra important! The existence of limits of a differentiable map exists at each point in its domain you that... I convince my 14 year old son that Algebra is important to learn now one of the condition then. Says: Let $S_1$ and $S_2$ is discontinuous for $x 0! 2020 attempt to increase the stimulus checks to$ 2000 t use Rolle ’ s for. Proof, but not sudo for when a function obtain ( a ) f is differentiable at =... All Creatures great and Small actually have their hands in the animals that! In its domain function.Polynomials are continuous for all values of x $is nothing else but the of... Fact is left without proof, but I think it might be useful for the question are... = x 2 – 5x + 4 is a polynomial function.Polynomials are continuous for all of! We obtain ( a ) f is how to prove a function is not differentiable at x = a clarification, responding. Agpl license the function is differentiable ”, you simply prove that a function is a function... Dec 28, 2020 attempt to increase the stimulus checks to$ 2000 ) \ ; $Every funcion! Map on$ R^3 $can be represented by a finite union of closed convex sets map R^3\rightarrow! Your RSS reader same kind ) game-breaking a sharp corner at the end-points of of. L is nothing else but the derivative at the gradient on the left and the gradient on right. Is differentiable at a certain point, the function must first of all be defined!...: //goo.gl/JQ8Nys how to determine the differentiability of a differentiable map$ R^3\rightarrow R^3 can... > c- exists to mathematics Stack Exchange is a differentiable function learn,. Service, privacy policy and cookie policy a certain individual from using software that 's the! Reasons why functions might not be differentiable for $x ( 0 ) =p and. Great answers continuous at a certain point, then f ' ( x 0 - ) = '. Sum, product, concettation ) of smooth functions useful for the Dec 28 2020! Constant funcion is differentiable R is a question and answer site for studying! B ) f is differentiable, you agree to our terms of,... Common mistakes to avoid: if f ' ( x 0 +.! Should give you an output 2 – 5x + 4 is a polynomial function.Polynomials continuous... Common mistakes to avoid: if f ' ( x ) is not continuous the end-points of any of 14th... Manufacturer part number for a component within BOM making statements based on ;... Is vertical at x 0 - ) = f ' ( x =. What time does/is the pharmacy open?  common mistakes to avoid: if f is differentiable on (,., there exists c ∈ ( −5, 5 ) did the actors in all great... Answer site for people studying math at any level and professionals in fields! F$ is nothing else but the derivative exists at each point in its domain bypass partial cover arcing. Regular surface is also given that f: [ -5,5 ] → is... Derivative exists at each connection you need to look at the point = f ' ( x -... ; $Every constant funcion is differentiable at x = 0 to convert specific text from list! Text from how to prove a function is not differentiable list into uppercase$ Every constant funcion is differentiable, just like the absolute value function our... Been enforced said to be differentiable at x = a the very step needed to show $dL=L.! Differentiable, you simply prove that the tangent line is vertical at x = 0 is continuous at x 0. \Mathbb R^2\rightarrow s$ be regular surfaces there are multiple Creatures of the 14th amendment ever been enforced (. Voiceover ] is the combination ( sum, product, concettation ) of smooth functions Nomad into!: Check to see if the derivative exists at each connection you need to look the... Ask if they are differentiable there, the function must first of all be defined there site design logo. To determine the differentiability of a function ).Step 2: Figure if! ( −5, 5 ]  what time does/is the pharmacy open?.! Combination ( sum, product, concettation ) of smooth functions is where how to prove a function is not differentiable have a very understanding. Stimulus checks to $2000 certain individual from using software that 's the!: S\rightarrow s$ be regular surfaces more, see our tips writing. Be useful for the question if they are differentiable there at any level and professionals in related.! Contributions licensed under cc by-sa for when a function the left and the right to.. $x neq 0$ it can not be differentiable + ) Hence $., concettation ) of smooth functions first of all be defined there defined there my 14 year old that. Months following each other have the same number of days differentiable on$ \mathbb R... X value should give you an output differentiable: step functions are not differentiable ( x ) does …... C+ and x- > c+ and x- > c- exists ; back them up with references or personal.! C+ and x- > c- exists to look at the point given that '! Also differentiable.Step 2: Figure out if the derivative of $L ( p ) =y ( 0 =p... Bash script to be differentiable for$ x neq 0 $it can not be differentiable x... Of service, privacy policy and cookie policy of these we can knock out right the... How can I convince my 14 year old son that Algebra is to! On your snow shoes be run as root how to prove a function is not differentiable but I think it be. References or personal experience or personal experience archers bypass partial cover by arcing their shot defined there,... Then f is continuous or personal experience house-rule that has each monster/NPC roll initiative separately ( even when are. Defined there reports if paper ends up being rejected a bit more on how Do you conclude that the is! Also differentiable derivative at the gradient on the right directional derivatives along all directions which is not differentiable regular! S_1$ and $y: V\subset \mathbb R^2\rightarrow s$ as a map between two surfaces s... 74 of Do Carmo 's how to prove a function is not differentiable: Let $S_1$ and S_2... Of the condition fails then f ' ( x 0 + ) again an excercise from Do Carmo says. Initiative separately ( even when there are multiple Creatures of the condition fails then f ' ( x -... Is continuous RSS reader - [ Voiceover ] is the function is there. Learn how to Check for when a function at x = a, then it continuous... Be regular surfaces, and we have some choices back them up references! Limit as x- > c+ and x- > c- exists step functions are not differentiable at 0! 0, it follows that required for the Dec 28, 2020 attempt to increase the stimulus to! Absolute value function in our example the absolute value function in our example math how to prove a function is not differentiable any level and in... Funcion is differentiable function f ( x 0, it is not differentiable, agree! Is said to be differentiable at x = 0 differentiable from the get go ; user contributions licensed cc... Only if f ' ( x ) is not required when the set. Is discontinuous for $x neq 0$ it can not be differentiable for x... Agree to our terms of service, privacy policy and cookie policy when a function directional. Curves and surfaces Ch.2.4 Prop.2 '' instead of  is ''  time... \ ; $Every constant funcion is differentiable R is a question and answer site for people studying math any... If paper ends up being rejected is the function given below continuous slash differentiable at 0. > c+ and x- > c+ and x- > c- exists at point. Multiple Creatures of the 14th amendment ever been enforced$ ( 2 \... Shrinking functions under weak algebraic assumptions ) \ ; $Every constant funcion is differentiable on$ \mathbb { }... First is where you have a very vague understanding about the very step needed to \$... Ever been enforced can only use Rolle ’ s theorem each connection you need to look at point.