# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### Finding the values of $a$ and $b$ such that $f$ is continuous and differentiable at $x = 1$? [closed]

The equation is $F(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ ax+b & \text{if } x>1 \end{cases}$ Differentiable at $x = 1$ I'm having a hard time understanding on how to ...
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### 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 ...
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### Does differentiability imply continuity for a derivative? [closed]

If $f(x)$ is differentiable at a point $c$, then is $f'(x)$ continuous at $c$? If so (or if not,) please provide a proof or a counterexample.
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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) ...
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### 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 $x=c$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\lim_{x \to b-}\dfrac{f'(x)}{g'(x)}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 f(x) = ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...