5
votes
3answers
131 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
1
vote
1answer
37 views

Continuous function with continuous one-sided derivative

Simple example of the absolute value function $x \mapsto |x|$ on $\mathbb{R}$ shows that it is possible for a continuous function to posses both the right-hand and the left-hand side derivatives and ...
0
votes
1answer
43 views

Extremely challenging differentiable function

If f is a differentiable function with $f(8) = 1$ and $f'(8) = 0$, find \begin{equation} \frac{d}{dx}\left({\sqrt{(x^2+x+3)f(x^3)}f(x^3)^2}\right) \end{equation} for $x=2$. I really really really ...
1
vote
1answer
40 views

Proving convexity of a function whose Hessian is positive semidefinite over a convex set

C is a convex set in R^n and f:R^n --> R is twice continuously differentiable over C. The Hessian of f is positive semidefinite over C, and I want to show that f is therefore a convex function. I ...
1
vote
1answer
62 views

partial derivative of $f(x,y)$ who satisfies $f_{xx}-f_{yy}=0$

Suppose that $z=f(x,y)$ and its second-order partial derivative is continuous. It also satisfies $\displaystyle\frac{\partial^{2}f}{\partial x^{2}}-\frac{\partial^{2}f}{\partial y^{2}}=0$,$f(x,2x)=x$ ...
0
votes
1answer
32 views

Differentiability of $f(x,y,z) = \sqrt{|xyz|}$

I need to study this function: $f(x,y,z) = \sqrt{|xyz|}$ at the point $P=(0,0,0)$. I have determined it is continuous at P. That is, $\lim_{(x,y,z)\to (0,0,0)} f(x,y,z) = f(0,0,0) = 0$ Also, that ...
0
votes
2answers
33 views

Prove the diffentiability of a function of a continous function at x=0.

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous at $x=0$, prove that $g(x)=xf(x)$ is differentiable at $x=0$. So this is the definition of a continuous function: $$\lim_{x \to 0} f(x) = f(0)$$ ...
3
votes
1answer
196 views

Finding continuity and differentiability of a multivariate function

Determine whether the following functions are differentiable, continuous, and whether its partial derivatives exists at point $(0,0)$: (a) $$f(x, y) = \sin x \sin(x + y) \sin(x − y)$$ ...
0
votes
1answer
26 views

If $f$ is $C^1(\mathbb{R})$, is it $C^1(\{a\})$?

Say I have a well-behaved function like $f(x)=x$. This is obviously $C^1$, but does it make sense to say the function is $C^1$ around a single point? A broader question, if $a\in\mathbb{R}$, does ...
3
votes
1answer
60 views

Suppose all partial derivatives of $f$ exist at $x_0$; is $f$ continuous at $x_0$?

Consider $f : C \to \mathbb{R}$ with $C \subset \mathbb{R}^n$ being open: Suppose $f$ is differentiable at $\mathbf{x}_0 \in C$. Is $f$ continuous at $\mathbf{x}_0$? Why? Suppose all partial ...
0
votes
4answers
58 views

If $f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$ and if $lim_{x\to\infty}f'(x)=0$ Then $f$ uniformly continuous on $[0,\infty)$

I got this problem: Let $f$ be a continuous function on $[0,\infty)$ and differentiable function on $(0,\infty)$ such that $\lim_{x\to\infty}f'(x)=0$. (1) Prove that for each $0<\epsilon$ there ...
0
votes
1answer
44 views

Continuity and differentiability on piecewise function

Let $$f(x)=\begin{cases}x^2-3, & x<0;\\-3, & x\geq 0.\end{cases}$$ (a) Find the value of $x$ where $f$ is discontinuous (b) Find the value of $x$ where $f$ is non-differentiable ...
2
votes
1answer
43 views

Derivative of an Inverse Function

Can someone please give me a simple proof of this- If $f$ is differentiable on an interval containing $c$ and $f'(c) \neq 0$, then $f^{-1}$ (inverse of $f$) is differentiable at $f(c)$. I can see ...
0
votes
1answer
46 views

Continuity and differentiablity [closed]

True or False ? If $f : \mathbb R \to \mathbb R$ satisfies $$|f(x) − f(y)| ≤ |x − y|^{\sqrt{2}}$$ for all $x, y \in R$, then $f$ must be a constant function. Let $f : \mathbb R\to \mathbb R$ be ...
-2
votes
1answer
67 views

derivability don't imply partial to be continuous ? example

Is $$f(x,y) =\begin{cases} x^2+2x+2y & \text{ for } (x,y)\neq (0,0) \\ y^2 & \text{ for } (x,y)=(0,0) \end{cases}$$ derivable? But its partials are not continuous?
0
votes
1answer
34 views

continuity, discontinuity derivative and relation to being derivative but its partials are not continuous

is $$f(x) =\begin{cases} x^2\sin(\frac{1}{x}) \mbox{ for } x\neq 0 \\ 0 \mbox{ for } x= 0\end{cases}$$ a continuous function specially at point x=0? And why being derivable its derivative is not ...
0
votes
0answers
24 views

Applications of Continuity and Differentiability on a Tough Qn

Given f is cont on [0,1] and that it is twice differentiable on (0,1). Suppose that Integral from 0 to 1 of f(x) dx = f(0) = f(1). Prove that there exist a number c where c is an element of (0,1) ...
3
votes
4answers
134 views

Derivability of a piecewise function

Let's say I have a continuous piecewise function of a single variable, so that $y = f(x)$ if $x < c$ and $y = g(x)$ if $x>=c$. Is it right to say that the derivative of the function at $x=c$ ...
0
votes
1answer
41 views

ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
0
votes
1answer
35 views

$f$ differentiable on $[a,b]$, but not Lipschitz

Question 11-37(d) of Spivak's Calculus, 4th ed., asks If $f$ is differentiable on $[a,b]$, is $f$ Lipschitz of order $1$ on $[a,b]$? The phrase "differentiable on $[a,b]$" is a little ...
0
votes
0answers
38 views

question on differentiable and continious function

How should the function $f(x)=x\operatorname{sgn} x$ be defined at $x=0$ so that it is continuous there? Is it then also differentiable? How should the function $g(x)=x^2 \operatorname{sgn} x$ be ...
0
votes
0answers
32 views

Covering up discontinuities to create analyticity

The floor function, $\lfloor x \rfloor$ , has a "jump" at the integers where its derivative ceases to exist. Everywhere else, its derivative is zero. Now, I wish to multiply the floor function by ...
2
votes
2answers
55 views

Show that the graph of $y=x^3\sin(\pi/x)$ extends to a smooth arc

Here's the problem: Let $y(x)$ be a real-valued function defined on the interval $x\in [0,1]$ by means of the equation $$y(x)= \left\{ \begin{array}{lr} x^3\sin(\frac{\pi}{x}) ...
3
votes
0answers
52 views

How many continuous functions are differentiable? [duplicate]

Consider the set of continuous functions $\mathbb{R} \to \mathbb{R}$. I assume that the subset that are not everywhere differentiable accounts for almost all of them. Is this true? What is the precise ...
8
votes
1answer
537 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
0
votes
1answer
54 views

Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$ where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$ Since it seems like, ...
0
votes
2answers
46 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
27 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
24 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
111 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
105 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
55 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
54 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
40 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 ...
1
vote
2answers
58 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
1answer
40 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 ...
6
votes
1answer
275 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
39 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
292 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
36 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
31 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
68 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
23 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
62 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 when $f$ is a piecewise defined function? A concrete example is below, but I'm interested in general ...
5
votes
1answer
108 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
93 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
21 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
417 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
65 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
168 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 ...