0
votes
2answers
39 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
2
votes
1answer
21 views

if $f([a,b])=[c,d]$ and $[c,d] \subset [a,b]$, is there $x \in [c,d]$ such that $f(x)=x$?

I'm trying to prove something that I'm not sure is correct. Let $f$ be a continuous, differentiable and monotonic function $f:[a,b] \to [c,d]$, where $[c,d] \subset [a,b]$. Is there an $x \in [c,d]$ ...
0
votes
1answer
20 views

Finding all continuity and differentiability points of a function

Let $$f(x) = \begin{cases} x^2(x^2-1),&x \in\mathbb{Q} \\ 0,&x \not\in\mathbb{Q} \end{cases}$$ A. When is this function continuous? when is it differentiable? I solved these kind of ...
1
vote
1answer
109 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
0
votes
1answer
102 views

$f(x,y) = (1- \cos(\frac{x^2}{y})) \sqrt{x^2+y^2}$

Let $f(x,y) = (1- \cos(\frac{x^2}{y})) \sqrt{x^2+y^2}$ for $y \ne 0$ How can I prove that f is not differentiable in $(0,0)$. Please some help.
2
votes
1answer
47 views

Modifications of Weierstrass's continuous, nowhere differentiable functions

Recalling how nowhere continuous functions such as the Dirichlet function can sometimes be modified on a $\lambda$-null set of points (in this instance, a countable set) to become everywhere ...
0
votes
1answer
43 views

Construct a continuous function which has no derivative almost everywhere.

Georg Cantor is famous for the first set theory (in "naive" terms) and the diagonal argument. However Cantor is also credited with the Cantor Set and for constructing a continuous function which has ...
2
votes
0answers
35 views

On continuous functions and second derivative

Let $f:[a,b]\to\mathbb R$ be a continuous function suh that $f''(x)$ exists $\forall x\in(a,b)$ . If $a<c<b$ and $f(a)=f(b)=0$ , then how to show that $\exists d\in(a,b)$ such that $f(c)=\dfrac ...
0
votes
0answers
26 views

Continuity and differentiability of a function. [duplicate]

How to prove that $f(x)$ is discontinuous at $x=0$ ? $$f(x)=\begin{cases} \sin\left(\tfrac1x\right),& \text{when $x\neq0$} \\ 0, & \text{when $x=0$} \end{cases}.$$
1
vote
2answers
30 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
0
votes
0answers
27 views

Questions from a calculus assignment about a function [duplicate]

Can anyone guide me through this problem? Let $f(x) = \lvert 4-x^2 \rvert$, $-4\le x\le 1$. Sketch (I have completed this part). Rewrite $f$ as a piecewise function. Give the range of $f(x)$. Use ...
0
votes
1answer
38 views

If a differentiable function has bounded derivative, Must it be that its derivative continuous?

I got this question: Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, If $f'$ is bounded on $(a,b)$, Must it be the case that $f'$ is ...
3
votes
0answers
87 views

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
1
vote
1answer
38 views

Question about limits and Mean Value Theorem

Let $f:(a,b) \rightarrow \mathbb{R}$ and $g:(a,b) \rightarrow \mathbb{R}$ be differentiable on (a,b) with $g'(x) \neq 0$ for all $x$ in $(a,b)$. Suppose $\lim_{x \to b-}\dfrac{f'(x)}{g'(x)}$ ...
8
votes
1answer
245 views

Does there exist a continuously differentiable function with the following properties?

Does there exist a continuously differentiable function $f: [1,5] \rightarrow \mathbb{R}$, such that $f(1) \lt 0, f(5) \gt 3$ and $f'(x) \leq e^{-f(x)}$? Now do I just integrate it to get $f(x) ...
1
vote
1answer
24 views

Can someone explain the concept of continuity and differentiability for functions of several variables?

Can someone explain the concept of continuity and differentiability for functions of several variables? Illustrated examples will definitely help, on how to solve problems(or establish proofs) of the ...
1
vote
0answers
28 views

Suppose limit as x approaches 0 of the derivative of f(x)=a. Show that the derivative at 0 exists and it is equal to a. [duplicate]

Let f be a continuous function on the interval [0,1], which is differentiable on (0,1). I know that I will have to use the definition of a derivative, which is the limit as x approaches 0 ...
4
votes
3answers
65 views

Find a continuous function on the reals where $f(x) >0$ and $f'(x) < 0$ and $f''(x) < 0$

We need to find a function $f(x)$ where $f(x) >0 $and $f'(x) < 0$ and $f''(x) < 0$ where $f$ is continuous for all real numbers. We have tried $ f(x) = \sqrt{-x}$ however this is not defined ...
1
vote
1answer
21 views

Statements about differentiation

Let $f:(a,b] \rightarrow \mathbb{R}$ be continuous on [a,b] and differentiable on (a,b). Suppose the limit $\lim_{x \to b-}f'(x)$ exists.Prove $f$ is differentiable from the left at $b$. Now ...
0
votes
1answer
53 views

How to find if and where $f(x)$ is continuous and/or differentiable for a given piecewise function? [closed]

What approach would be ideal in finding if and where $f(x)$ is continuous and/or differentiable for $ f(x) = \left\{ \begin{array}{lr} ln(x) & : x > e\\ \frac{x}{e} & : x \leq e ...
5
votes
1answer
88 views

Is the standard part function another devil's staircase?

The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means). I was incredibly amazed when I found out ...
2
votes
2answers
88 views

Properties of the function defined by $g(x) = \sum\limits_{n=0}^{\infty} \frac{1}{1+n^2x^2}$

I am looking at the function $g:\mathbb{R} \rightarrow \mathbb{R}$ defined as $$g(x) = \sum\limits_{n=0}^{\infty} \frac{1}{1+n^2x^2}$$ I would like to know if this function is convergent, continuous ...
0
votes
1answer
19 views

Proving a function is bounded above.

Hi all, while doing this question ,I feel that I understand the concept of the question, but can't seem to formulate it into a viable answer. If the limit as $x \rightarrow \infty$ is the same as $x ...
6
votes
4answers
402 views

Construct a function that is nowhere differentiable.

I have been working on this question for a very long time now and seem to have reached a dead end, I will show all my attempted solutions, and any help on the various parts of the question would be ...
4
votes
1answer
45 views

Differentiability implies continuity — possibly pedantic question about the common proof

The common proof that differentiability implies continuity arrives at this limit: $$\lim_{x\to a} [f(x) - f(a)] = 0$$ I'm failing to see the simple justification for moving to the next step, which ...
5
votes
1answer
156 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
71 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
26 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
70 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
17 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
48 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
23 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
68 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
56 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
68 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
51 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
96 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
57 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
38 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
42 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
67 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
71 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
34 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
37 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
43 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 ...
4
votes
1answer
454 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
49 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
49 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
74 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
27 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