4
votes
1answer
99 views

Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
2
votes
1answer
59 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
0
votes
1answer
23 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...
0
votes
2answers
65 views

Derivative of $x^2\sin(\frac{1}{x})$

I was reading an article in American Mathematical Monthly and came across this example.It says that derivative of $x^2\sin(\frac{1}{x})$ takes on all values in $[-1,1]$ in any interval ...
0
votes
1answer
14 views

Differentiability and basic definitions

If $f+g$ is differentiable at $a$, must $f$ and $g$ be differentiable at $a$? If " and $f$ is differentiable at $a$, must $g$ be differentiable at $a$? If $f*g$ is differentiable at $a$ and $f$ is ...
3
votes
2answers
44 views

Examples of Functions

Alright so I am trying to find examples of functions that are differentiable at a point, but not continuous there. Also a function continuous at no point; a function continuous only at one point. ...
0
votes
1answer
19 views

Continuity of a partial derivative

I have the function $$f(x,y)=\begin{cases} x^2ysin(\frac1x) & \text{if $x$ is not 0} \\ 0 & \text{if $x=0$}\end{cases}$$ And I need to find the derivative and the ...
1
vote
3answers
43 views

If Limit of function and derivative exist, then limit of derivative is 0 [duplicate]

Any hints for this question , My attempt; Say $f(x):0$$\rightarrow$$\mathbb{R}$ The by MVT, there exists a $c$$\in$$(0,\infty)$ , such that; $f'(c)=$$\frac{f(x)-f(0)}{x-0}$ but im not sure about this ...
0
votes
1answer
46 views

Differentiability conditions for a piecewise function

So this is an analysis class, and we just started the unit on differentiability -- however I missed the class. Can someone start me off with a good real analysis definition for differentiability of ...
2
votes
1answer
46 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
1
vote
2answers
43 views

What is the difference between “differentiable” and “continuous”

I have always treated them as the same thing. But recently, some people have told me that the two terms are different. So now I am wondering, What is the difference between "differentiable" and ...
1
vote
1answer
66 views

Show a function is not continuous at a point

$$ f(x,y) = \begin{cases} \dfrac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y)\neq(0,0) \\ 0, & \text{if }(x,y)=(0,0) \end{cases} $$ For the definition of differentiability, I have: $$\lim_{h ...
0
votes
0answers
35 views

Totally differentiable function - definition

I know for a function of several variables, if all partial derivatives exist and they are continuous at and around a point $a$ then the function is totally differentiable at that point. I ...
1
vote
2answers
32 views

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$)

Let $f$ be a continuous function on an interval around $0$ and let $a_i=f(\frac{1}{i})$ (for large enough $i$) i) Suppose $\sum a_i$ converges. Must $f'(0)$ exist? ii) Suppose $f(0) = f'(0) = 0$. ...
0
votes
0answers
39 views

Seemingly easy analysis problem but unsure how to proceed.

if $f(x)=\frac{1}{x+2}$ then $f(x)=1-(x+1)+(x+1)^2+T$ for some $x_0$ between $x$ and $-1$ where $T=-\frac{(x+1)^3}{(2+x_0)^4}$ I'm not sure how to proceed in solving this problem. We recently ...
1
vote
1answer
54 views

Bounded Jacobian implies uniform continuity

I am trying to solve the following problems but I am not sure what the difference between the 2 problems is. 1) Prove that is $U = B_r(x)$ (open ball centered at $x$ with radius $r>0$) is an open ...
-1
votes
1answer
40 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
0
votes
1answer
33 views

Prove something that is differentiable

The question states If g(x) is differentiable, then for any positive integer $n$, $(g(x))^n$ is differentiable and $\frac d{dx}$$(g(x))^n=(g(x))^{n-1}g'(x). $ Where does the continuity of g enter ...
0
votes
0answers
32 views

Confusion about validity of integration by substitution method

Suppose $$ f(x,y) = \int g(x,y,t)\, dt, $$ and I wish to do the integral over $ t $ by setting $$ t = h(x,y,\tau). $$ As an example, suppose $$ h(x,y,\tau) = \frac{\tau}{\cos x} -\tan y. $$ If we ...
0
votes
1answer
24 views

Differentiability of trigonometric piecewise functions

So I have a function of a real variable $x$: $f(x) = \left\{\begin{array}{lr} x \int_0^{tanx} \dfrac{t^2}{\sqrt{1+t^3}}dt & if \: x \ge 0\\ sin^2(x) & if \: x \lt 0 ...
3
votes
1answer
271 views

If $f$ is a twice differentiable function and $f(2^{-n}) = 0 $, for all $n \in \mathbb N$, then $f^\prime(0) = f^{\prime\prime}(0) = 0$.

Let $f : \mathbb R \to \mathbb R$ be a twice differentiable function, such that $f(2^{-n}) = 0$, for all $n \in \mathbb N$ . Show that $$f^\prime(0) = f^{\prime\prime}(0) = 0.$$ My attempt. First, ...
0
votes
1answer
42 views

Differentiability with non continuous partials (origin)

The function $$f(x,y)= \frac{x^{2}y^{2}}{(x^{2}+y^{4})} \quad if \quad (x,y) \neq (0,0)$$ $$f(0,0)=0$$ In order to study it's differenciability at the origin, I've studied if the partial are ...
1
vote
0answers
42 views

Continuity and differentiability of $x^a\sin ({1\over x}) $ at $0$

Consider the function $$ g_a (x) = \begin{cases} x^a\sin ({1\over x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$ I am looking to determine for which $a$ the map $g_a$ is differentiable on ...
1
vote
2answers
73 views

Counterexample - Increasing function

My intuition says that the statement is false. Anyone out there know of counterexamples? Suppose $f: R\to R$ and $c\in R$ such that $f'(c) > 0$. So, $\exists \varepsilon> 0 $ such that ...
0
votes
1answer
24 views

continuity with lipschitz dertivative

Can somone explain me what does it mean to say a "function is continously differentiable with a Lipschitz derivative near the limit point". I dont understand the technical jargoans. Thanks
2
votes
1answer
62 views

Proving nondifferentiability at all points of a continuous function

Given: $f_1(x)=x$ if $x\le1/2$ $f_1(x)=1-x$ if $1/2\le x\le1$ $f_1(x+1)=f_1(x)$ $\forall n\ge2,f_n(x)=(1/2)*f_{n-1}(2x)$ Let $S_m(x)=\sum_{n=1}^m f_n(x)$ $S_m$ is a continuous function on ...
0
votes
0answers
34 views

Lagrange MVT and finding function [duplicate]

I have two questions (use of IVT,MVT and derivatives is allowed, integrals are not allowed as well as Riemann integration): First question: Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable ...
0
votes
1answer
36 views

Proving this Taylor-esque expansion for a $C^2$ function vanishing at 0 and 1

I am trying to prove the following (which I think is true!): if $f:[0,1]\rightarrow \mathbb{R}$ is twice continuously differentiable and $f(0)=0=f(1)$, then for every $x \in (0,1)$ there exists $\xi ...
2
votes
2answers
163 views

Showing that the function is continuous but not differentiable

Let $$ f(x,y) = \begin{cases} \dfrac{xy}{\sqrt{x^2+y^2}} & \text{if $(x,y)\neq(0,0)$ } \\[2ex] 0 & \text{if $(x,y)=(0,0)$ } \\ \end{cases} $$ Show that this function is continuous but not ...
0
votes
1answer
45 views

Ensuring continuity and differentiability of a function

I'm totally stuck with this function of which I have to prove its continuity and differentiability: $$ f(x)=\begin{cases} a+\sqrt{x^2+3},& x\le 1,\\ b\ln x+(2a+1)x,& ...
0
votes
1answer
44 views

If $f(x)$ is differentiable on $(a,b)$, is $f(x)$ continuous on $[a,b]$?

I am wondering if this is true. I have seen some books which say that $f(x)$ is differentiable on $[a,b]$ instead of $(a,b)$, and thus avoid this problem.
-1
votes
3answers
264 views

Example of uniformly continuous function on R that is not differentiable on all of R

Give an example of a uniformly continuous function $g:\mathbb{R} \rightarrow \mathbb{R}$ that is not diff erentiable on all of $\mathbb{R}$. Hmm. I can't think creatively enough for one! Would f(x) = ...
0
votes
1answer
58 views

If $f(x)=\chi_{(0,\infty)}\exp(-1/x)$, show that $f\in C^{\infty}$.

Define the function $f:\mathbb{R}\to\mathbb{R}$ as follow: $f(x)=\chi_{(0,\infty)}\exp(-1/x)$ In other words: $f(x)=0$ if $x\le 0$, and $f(x)=\exp(-1/x)$ if $x>0$. Show that $f\in C^{\infty}$. ...
0
votes
1answer
37 views

How do you prove $e^{-a}=a$ without using graphs?

We're doing a section on limits, continuity, and differentiation in my Advanced Calculus class, and I am at a loss for how to prove this...
2
votes
1answer
44 views

Continuous but nowhere differentiable function on domain

I am trying to come up with an example of a function which is continuous on $[0,1]$ but nowhere differentiable but which is in a way simpler than Weierstrass function or "more intuitive" so to speak. ...
1
vote
2answers
74 views

Real Analysis Derivative

Consider the function $f\colon\mathbb R\to\mathbb R$ where $f$ is defined by $$f(x)= \begin{cases} x^b\sin(1/x), &\text{if $x>0$};\\ 0,&\text{if $x<=0$.} \end{cases}$$ Prove that the ...
0
votes
1answer
57 views

B-spline parameterization and derivatives

I have a question regarding the re-parameterisation of a B-spline. Some info: The B-spline is of order 4 (degree 5), hence $C^3$ continuity There is no knot multiplicity The end conditions are not ...
0
votes
3answers
96 views

Find all points where $f(x)$ fails to be differentiable. Justify your answer

Find all points where $f(x)$ fails to be differentiable. Justify your answer $$f(x) = |x| - 1$$ I am confused with continuity of it and cannot turn it into piecewise function and finding the limit ...
0
votes
1answer
51 views

Two $C^\infty$ functions which agree on a set containing an accumulation point, but do not agree on *any* neighborhood?

As I understand it, two analytic functions defined on $\mathbb{R}^k$ which agree on a set with an accumulation point must agree on a neighborhood; however, the same is not true of $C^\infty$ ...
1
vote
1answer
37 views

Just a question regarding continuous differentiability

$ f: [0,1] \to [0,1] $ be a MONOTONE & CONTINUOUS function. Does it always imply that: $ f(x) $ is continuously differentiable??
0
votes
1answer
705 views

The definition of locally Lipschitz

I am given this definition: A function $f:A\subset\mathbb R^n\to\mathbb R^m$ is locally Lipschitz if for each $x_0\in A$, there exist constants $M>0$ and $\delta_0 >0$ such that ...
0
votes
2answers
66 views

Is it possible to have a function differentiable but not continuous in a given interval?

Is there any possible function that is not continuous but differentiable in a given interval. It sounds non-logical to me since differentiation is a special limit function in itself therefore ...
1
vote
2answers
265 views

Piecewise interpolation with derivatives that is also twice differentiable

This question regards the issue of interpolation of one dimension real functions. If one has a finite set of function values and its corresponding derivatives, one could find unique continuous ...
1
vote
1answer
293 views

one-sided continuity and one-sided derivative?

A continuous function is continuous at an $x$ value (call the $x$ value that we're interested in $c$) if both of these conditions are met and are true: $f(c)= \text{some real number}$ ...
1
vote
1answer
189 views

Given a function has second derivative,find the unknown constant a.

I have done questions which asks to find unknown constants given the function is continuous. But, this question provides the existence of second derivative at x=1. What information should I draw ...
1
vote
1answer
126 views

Showing limit of a derivative is finite

Given that a function $f$ is continuous on interval$\left[a, b\right]$, and that its derivative is finite everywhere on that interval except possibly at $c$. I am also given that $lim_{x \rightarrow ...
3
votes
3answers
109 views

continuity of the derivative under certain conditions

I am working on this exercise in a book which asks to prove that $f$ is differentiable if $f$ is continuous and that $\lim \limits_{x\rightarrow x_0} f'(x)$ exists. I know that this is easy to show ...
0
votes
1answer
214 views

Derivative inequality for a twice continuously differentiable function.

This is a question from a past exam. I thought that this was easy, but found no way of solving it. Let $f: \mathbb R\rightarrow\mathbb R$ be continuously twice differentiable with ...
-2
votes
1answer
78 views

$f(x) = x^2$ for $\Bbb{Q}$(rational) and $3(x^2)-2$ for $\Bbb{R-Q}$(irrational).Check differentiability at $x=\pi$. [closed]

Let $f$ be the following function, $$ f(x) =\begin{cases} x^2 & x\in\Bbb{Q}\\ 3(x^2)-2 & x\in\Bbb{R\setminus Q}\end{cases}$$ Check differentiability at $x=\pi$. Please help soon.
3
votes
2answers
2k views

An inflection point where the second derivative doesn't exist?

A point $x=c$ is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point. And a list of possible inflection points will be those points ...