# Tagged Questions

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### Where is $f(x)=\sqrt{|1-x^2|}$ Lipschitz continous?

It seems to me that the Lipschitz constant is 1 near $x=\pm 1$, $y= \pm 1$ $$|f(x)-f(y)| \leq \frac{|x+y|}{\sqrt{|1-x^2|}+\sqrt{|1-y^2|}}|x-y|$$ How would you define the Lipschitz constant L?
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### What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
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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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### Why are the following graphs discontinuous at $f(0)$ (epsilon-delta)

The caption for graph (f) is "Infinite jump". The caption for graph (h) is "Infinitely many infinite jumps". The graphs are meant to illustrate that we can pick arbitrarily small intervals around ...
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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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### From essential oscillation to a continuous representative

Let $u$ be a measurable function such that for every(former: a.e.) $x\in \Omega$ there holds for sequences $R_n,\delta_n\to 0$ that$$\omega_n:=ess-osc_{B_{R_n}(x)} u\leq \delta_n. \tag{1}$$ Edit: ...
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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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### Lipschitz continuity power type function [duplicate]

Is the function $f(x)=x^{\gamma+1}$, where $x>0$ and $\gamma<0$ Lipschitz continuous ? I am a bit confused !
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