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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Lipschitz Continuous $\Rightarrow$ Uniformly Continuous

The Question: Prove that if a function $f$ defined on $S \subseteq \mathbb R$ is Lipschitz continuous then $f$ is uniformly continuous on $S$. Definition. A function $f$ defined on a set $S \subseteq \mathbb R$ is said to be Lipschitz continuous…
Moderat
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The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that $f$ is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So why in most of the textbooks they always mention…
ManiAm
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How do I show that all continuous periodic functions are bounded and uniform continuous?

A function $f:\mathbb{R}\to \mathbb{R}$ is periodic if there exits $p>0$ such that $f(x+P)=f(x)$ for all $x\in \mathbb{R}$. Show that every continuous periodic function is bounded and uniformly continuous. For boundedness, I first tried to show…
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Is there a bijective seacucumber?

A friend defined a seacucumber as a continuous function $f:\mathbb{C}\to\mathbb{C}$ such that $f(z+1)+f(z+i)+f(z-1)+f(z-i)=0$ for all $z\in\mathbb{C}$. He wanted to know if there exists a bijective seacucumber. An example of a non-trivial…
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Assume $f\left({\frac{f(x)}{x}}\right)= f(x)$. Show that $ f$ continuous

Let $f : (0, \infty) \to (0, \infty)$ be a function that has primitives (that is, there is $F$ so that $F' = f$) and satisfies the relation $$f\left(\displaystyle{\frac{f(x)}{x}}\right)= f(x) , \forall x \in (0, \infty)$$ Prove that $f$ is…
C_M
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Does there exist a space filling curve which sends every convex set to a convex set?

Does there exist a surjective continuous function $f:[0,1]\to [0,1]^2$ which maps every convex set to a convex set? Such a function could be considered an especially "regular" sort of space-filling curve. There are of course many well-known…
user456828
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Is there a monotonic function discontinuous over some dense set?

Can we construct a monotonic function $f : \mathbb{R} \to \mathbb{R}$ such that there is a dense set in some interval $(a,b)$ for which $f$ is discontinuous at all points in the dense set? What about a strictly monotonic function? My intuition…
A.S
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Can a continuous function from the reals to the reals assume each value an even number of times?

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. Is it possible for $f$ to assume each value in its range an even number of times? To clarify, some values might be taken 0 times, some 2, some 4, etc., but always an even (and therefore…
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Is there a function having a limit at every point while being nowhere continuous?

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ which has a limit at every $x\in\mathbb R$ and is everywhere discontinuous?
Curious Droid
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Prove that the inverse image of an open set is open

Let $ X \subset \mathbb{R}$ be a non-empty, open set and let $f: X \rightarrow \mathbb{R}$ be a continuous function. Show that the inverse image of an open set is open under f, i.e. show: If $M \subset \mathbb{R}$ is open, then $f^{-1}(M)$ is open…
eager2learn
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How can a function with a hole (removable discontinuity) equal a function with no hole?

I've done some research, and I'm hoping someone can check me. My question was this: Assume I have the function $f(x) = \frac{(x-3)(x+2)}{(x-3)}$, so it has removable discontinuity at $x = 3$. We remove that discontinuity with algebra: $f(x) =…
1Teaches2Learn
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How to solve $\dot{x} = \frac{f(x)}{\|f(x)\|}$?

How to solve the following ODE? $$\dot{x} = \frac{f(x)}{\|f(x)\|},$$ where $x : \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f : \mathbb{R}^n \to \mathbb{R}^n$ is a continuously differentiable function with…
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Why is continuity characterized with open sets?

Why is the topological definition of continuous in terms of open sets? I think my main complaint might be that the notion of open set seems too flexible/general and considers too many things that don't seem the right notion of "closeness".…
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Is the distance function in a metric space (uniformly) continuous?

Let $(X, d)$ be a metric space. Is the function $x\mapsto d(x, z)$ continuous? Is it uniformly continuous?
Renato
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Prove: bounded derivative if and only if uniform continuity

The definition of uniform continuity of a real-valued function states: A function $f\colon A\mapsto\mathbb{R}$ is uniformly continuous on $A$ iff for every $\varepsilon \gt 0$ there exists a $\delta \gt 0$ such that for every $x$ and $y$ in $A$,…
chharvey
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