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)

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $f : X \to Y$ is said to be continuous at $x_0$ if for every $\varepsilon > 0$, there is $\delta > 0$ such that $d_Y(f(x_0), f(x)) < \delta$ whenever $d_X(x_0, x) < \varepsilon$. A map $f : X \to Y$ is said to be continuous if it is continuous at $x_0$ for all $x_0 \in X$.

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A map $f : X \to Y$ is said to be continuous if $U \in \tau_Y$ implies that $f^{-1}(U) \in \tau_X$.

In the case of metric spaces, the metric induces a topology, and the two notions of continuity coincide. Note that multiple metrics can induce the same topology, and that not all topologies are metrizable (can be generated from some metric).

Continuity is a sufficient condition for the intermediate value theore. It is necessary for the extreme value theorem, as well as differentiability.

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Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
AndrePoole
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Prove that the function$\ f(x)=\sin(x^2)$ is not uniformly continuous on the domain $\mathbb{R}$.

If I want to prove that the function$\ f(x)=\sin(x^2)$ is not uniformly continuous on the domain $\mathbb{R}$, I need to show that: $\exists\varepsilon>0$ $\forall\delta>0$ $\exists{x,y}\in\mathbb{R}\ : |x-y|<\delta$ and $|\sin(x^2) -…
Jeroen
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What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference geometrically? What is the best way to describe the difference between these two concepts? Where the motivation of Uniform Continuity came from? Thank You.
Bumblebee
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Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space.

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}\mid f \text{ differentiable with }f' \text{ continuous}\}$$ with the $C^1$-norm $$\lVert f\rVert := \sup_{a\leq x\leq b}|f(x)|+\sup_{a\leq…
RDizzl3
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Is an integral always continuous?

Say I have a function $f(x)$ on some interval $[a,b]$. Say it is integrable such that $\displaystyle\int f(x)~dx $ is defined. Is $\displaystyle\int f(x)~dx $ necessarily continuous? If I were to know that the integral is integrable itself such…
vondip
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Why is this combination of nearest-integer functions --- surprisingly --- continuous?

Alright, I didn't know the best way to formulate my question. Basically, whilst doing some physics research, I naturally came upon the function $$ f(x) = 2x[x] - [x]^2 $$ where I use $[x]$ as notation for the `nearest-integer function' (i.e.…
Ruben Verresen
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Why weren't continuous functions defined as Darboux functions?

When we were in primary school, teachers showed us graphs of 'continuous' functions and said something like "Continuous functions are those you can draw without lifting your pen" With this in mind I remember thinking (something along the lines…
YoTengoUnLCD
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Is there another topology on $\mathbb{R}$ that gives the same continuous functions from $\mathbb{R}$ to $\mathbb{R}$?

If we look at any set $X$ with the trivial topology, then all functions from $X$ to $X$ are continuous. We could also take the discrete topology and get the same result: all functions are continuous. Another example: Take the Sierpinski space, all…
Jens Renders
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Can $ f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $f^{-1}(y) = \{a_y,b_y\} $ be continuous?

This is the problem we want to solve: Can $f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $ f^{-1}(y) = \{a_y,b_y\}, a_y \neq b_y $ be continuous? Originally I've seen this question on an exam but it was…
Ormi
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Why are removable discontinuities even discontinuities at all?

If I have, for example, the function $$f(x)=\frac{x^2+x-6}{x-2}$$ there will be a removable discontinuity at $x=2$, yes? Why does this discontinuity exist at all if the function can be simplified to $f(x)=x+3$? I suppose the answer is that you can't…
Kyle Delaney
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Do all continuous real-valued functions determine the topology?

Let $X$ be a topological space. If I know all the continuous functions from $X$ to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is somewhat artificial. So if this is wrong, will it be right if $X$ is a topological…
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How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

Prove that there is no continuous functions $f:[0,1]\to R$, such that $$ f(x)+f(x^2)=x. $$ My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all $x\in…
math110
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Equicontinuity on a compact metric space turns pointwise to uniform convergence

I know that If $\{f_n\}$ is an equicontinuous sequence, defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly. I'm having trouble proving this. I see the same problem here but am…
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Proof of continuity of Thomae Function at irrationals.

In Thomae's Function: $$ \begin{align} t(x) = \begin{cases} 0 & \text{if $x$ is irrational}\\ \frac{1}{n} & \text{if $x = \frac{m}{n}$ where $\gcd(m,n) = 1$} \end{cases} \end{align} $$ I can prove the discontinuity at rational $b$ by taking a…
Alvis
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Additivity + Measurability $\implies$ Continuity

A function $f:\Bbb R \to \Bbb R$ is additive and Lebesgue measurable. Prove that $f$ is continuous. I know that on $\Bbb Q$, $f$ comes out to be linear. So, if $f$ is to be continuous then $f$ must be linear in $\Bbb R$. But, I'm stuck here. If…
Aang
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