# Tagged Questions

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### Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
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### Solution of differential equations with discontinuity

Suppose that we have scalar differential equation $$\dot{x}(t)=u(t)$$ Here $u(t)$ is a piecewise constant function with discontinuity. If the points of discontinuity is ...
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### Is $||u||_{C^\alpha} \leq ||u||_{C^1}$ for all $u$?

We have $||u||_{C^\alpha,\Omega} = \text{sup}_\Omega |u(x)|+ \text{sup}_\Omega \frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $||u||_{C^1} =\text{sup}_\Omega |u(x)| + \text{sup}_\Omega|\frac{du}{dx}|$ I have ...
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### Continuity along different spaces

1) Say I have a function that is continuous along $\mathbb{R}.$ Would that function be then continuous along $\mathbb{Q}$ ? How about the other way around? 2) If I have two functions that are not ...
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### Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
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### Does differentiability imply absolute continuity? [duplicate]

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a function which is (i) differentiable at all $x \in (a,b)$ (ii) the right-derivative at $x=a$ exists and the left-derivative at $x=b$ exists. Does it ...
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### $\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0$ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
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### $f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant.

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that $$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.
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### Continuity of function does not imply continuity of extension

Let $f$ be increasing on a dense subset $D$ of $\mathbb{R}$, and define $\tilde{f}$ on $x\in\mathbb{R}$ $\tilde{f}(x):=\inf_{x<t\in D}f(t)$. Show that the continuity of $f$ on $D$ does not imply ...
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### Existence of increasing, smooth modulus of continuity

First, recall the definition: Given a function $f:M\to N$, where $M$ and $N$ are metric spaces, a modulus of continuity for $f$ is a function $\omega:[0,\infty)\to[0,\infty)$ such that ...
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### About maximum function and continuity

Let $\bar{x}\in\mathbb{R}^n$, $R>0$, and $P$ metric space. If $f:\bar{B}(\bar{x},R)\times P\rightarrow\mathbb{R}$ is a continuous function. We define $F:P\rightarrow\mathbb{R}$ by ...
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### Show that $f*(x) = \sup \{ f(y) : a \leq y \leq x \}$ is a non-decreasing continuous function

I am currently working on a problem and stuck on it. Here is the problem (it comes form Elementary analysis, the theory of Calculus by K. Ross P.153): Q: Let $f$ be a continuous function on [a,b]. ...
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### Prove there exists a point $c$ such thst $f(c)=c$ for the following function

If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function with $f(0)=2$ and $|f'(x)| \leq 1/2$ for all $x$ then there is a point $c$ such that $f(c)=c$ . My Attempt Let ...
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### A basic question about upper hemicontinuity

Given a correspondence $f:X\rightarrow 2^X$, suppose X is a closed simplex in $\mathbb{R}^n$, and $f$ is compact-valued. We say $f$ is upper hemicontinuous if, $\forall x\in X$ and every open subset ...
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### Example of continuous function without fixed point.

I need to find an example of a continuous function without a fixed point, and this is what I've come up with: As {1} is not in the (co)domain, I can evade all $x$ for which $f(x)=x$ up until I ...
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### Prove the following statements

Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous with $f(0)=0$ and $f(1)=1$. For the following you may apply standard results without proof provided you state them carefully; $(1)$ If ...
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### If $g$ is continuous and $f$ is s.t $f=g$ for $|x|<1$ then $f$ is continuous at 0

If $g:\mathbb{R} \rightarrow \mathbb{R}$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ is such that $f(x)=g(x)$ for all $|x|<1$ then $f$ is continuous at 0. Attempt; My claim is this statement ...
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### If $f,g$ are uniformly continuous prove $f+g,fg$ are uniformly continuous

Suppose $f:E \rightarrow \mathbb{R}$ and $g:E \rightarrow \mathbb{R}$ are uniformly continuous, where $E$ is a subset of $\mathbb{R}$. Show that $f+g \ \ and \ \ fg$ are uniformly contiuous, what ...