Questions tagged [continuity]

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)

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What functions can be made continuous by “mixing up their domain”?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ so that $f\circ \phi$ is continuous. So one could say a potentially ...
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How to prove using elementary methods that this function is everywhere continuous but nowhere differentiable?

Let $f$ be the function defined on all of $\mathbb{R}$ by the formula $$f(x) \colon= \sum_{n=0}^\infty \frac{1}{2^n} \cos \left( 3^n x \right).$$ How to show (rigorously but through elementary ...
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A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ...
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Is there any continuous function that is only differentiable on $\mathbb{Q}$?

I am looking for a continuous function $f: \mathbb R \rightarrow \mathbb R$ so that $f$ is differentiable in $x$, if and only if $x \in \mathbb Q$. I already know there is no function that is ...
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Mistake in proof of $\overline{C_c(X)} = C_0(X)$..

Usually one employs Urysohn's Lemma to proof the above identity. See for example here in the first answer: Show that the closure of $C_c(X)$ is $C_0(X)$. I find that proof rather complicated ...
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Under which circumstances is the space of Hölder continuous functions a Banach space?

Let $(M,d)$ be a metric space $\Lambda\subseteq M$ $E$ be a $\mathbb R$-Banach space Let $$\left\|f\right\|_{B(\Lambda,\:E)}:=\sup_{x\in\Lambda}\left\|f(x)\right\|_E\;\;\;\text{for }f:\Lambda\to E.$$...
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Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ is ...
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Differences between a quotient map and a continuous function in topology

Def. for a continuous function: Let $X$ and $Y$ be topological spaces. A function $f : X \rightarrow Y$ is continuous if $f^{-1} (V)$ is open in $X$ for every open set $V \subseteq Y$. Def. ...
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$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...
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Differentiability-Related Condition that Implies Continuity

I previously asked a related question here that I did not phrase as I intended. This is a revision of that question: It is a well-known fact that differentiability implies continuity. And, for ...
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Functions in a Reproducing Kernel Hilbert Space are Lipschitz continuous

I would like to show that all the functions in a Reproducing Kernel Hilbert Space (RKHS) are Lipschitz continuous. So that, I take two points in the domain $\vec{x}_{1} ,\vec{x}_{2} \in X$ then from ...
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Why is this Takagi's function continuous?

1903 Takagi constructed the function $f: [0,1] \rightarrow \mathbb{R}$ with $f(x) := \sum_{k=0}^\infty 2^{-k} \mathrm{dist}(2^k x, \mathbb{Z})$ where $\mathrm{dist}(x,A) := \inf\{|x-y| : y \in A\}$ ...