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) ...

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A Topology such that the continuous functions are exactly the polynomials

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the ...
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Why are real numbers useful?

A question (by a fellow CS student taking a first course in calculus, presumably after the lecture in which continuity was introduced: was as follows. In the real, physical world, we deal with ...
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Why did mathematicians introduce the concept of uniform continuity?

I have solved many problems regarding uniform continuity, but still I can't understand the following: Is there any practical application of this concept, or it is just a theoretical concept? Is there ...
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A game with $\delta$, $\epsilon$ and uniform continuity.

UPDATE: Bounty awarded, but it is still shady about what f) is. In Makarov's Selected Problems in Real Analysis there's this challenging problem: Describe the set of functions $f: \mathbb R ...
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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 ...
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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
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What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
22
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No continuous function switches $\mathbb{Q}$ and the irrationals

Is there a way to prove the following result using connectedness? Result: Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} ...
20
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How does this discontinuity occur in evaluating a nested square root?

This question is based on a comment I made on a question likely to be closed. Let $$y=\sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \dots}}}}}$$ be the classic nested square root which has ...
19
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2answers
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Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
19
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Which $f$ satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...
19
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Function $f(x)=\int_0^\infty\left|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\right|\,dt$

Let $$f(x)=\int_0^\infty\Big|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\Big|\,dt,$$ where $|\dots|$ denotes the absolute value. We are concerned only with positive values of $x$ (i.e. let the domain of the ...
19
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1answer
495 views

Is there a function having a limit at every point while being nowhere continuous?

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?
18
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1answer
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$f(f(\sqrt{2}))=\sqrt{2}$ then f has a fixed point

$f(x)$ is continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ $f(f(\sqrt{2}))=\sqrt{2}$ Prove that $f$ has a fixed point in other words prove the there is $x_1$ such that $f(x_1)=x_1$ I tried using ...
18
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A topological function with only removable discontinuities

I've posted similar questions here and here, but no one has answered them to my satisfaction. Suppose that $f:\mathbb{R} \to \mathbb{R}$ is such that $\lim_{y\to x}f(y)$ exists for all $x$, that is, ...
18
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1answer
356 views

Calculus over $\mathbb{Q}$

The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust ...
17
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6answers
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If $f$ is continuous at $a$, is it continuous in some open interval around $a$?

If $f: \mathbb{R} \to \mathbb{R}$ is continuous at $a$, is it continuous in some open interval around $a$?
17
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1answer
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Is It Always Possible to Draw A Connected Compact Set in $\mathbb R^2$?

Inspired by this answer, I wondered whether a printer could render all continuous functions "well enough". In particular, I am curious about the following statement: Let $S$ be a compact, ...
16
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6answers
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How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity?

I'm told that a function defined on an interval $[a,b]$ or $(a,b)$ is uniformly continuous if for each $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that $|x-t|\lt \delta$ implies that ...
15
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Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
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How to show that $f(x)=x^2$ is continuous at $x=1$?

How to show that $f(x)=x^2$ is continuous at $x=1$?
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1answer
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Prove that function is continuous without knowing the function explicitly

Let $f\colon \mathbb R^+\to\mathbb R$ be a function that satisfies the following conditions: $$\tag1 \lim_{x\to 1}f(x)=0 $$ $$\tag2f(x_1)+f(x_2)=f(x_1x_2)$$ Show that $f$ is continuous in its domain. ...
14
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If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $[0,1]$?

This is a self-posed question, so I do not know the answer and I would like to know what do you think about. Let $f,f_n:[0,1]\to \mathbb R$ be continuous functions. Assume that for every sequence ...
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Who gave you the epsilon?

Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his ...
14
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217 views

Help me correct my ideas of continuity

I've been studying real analysis over the past few months, and I'm having trouble organizing the different notions of continuity and ideas related to continuity in my head geometrically. I will ...
14
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1answer
377 views

Condition for an additive function to be continuous

The problem below is Problem 7 from this year's Miklos Schweitzer competition (contest ended Nov 4th). Suppose that $f: \Bbb{R} \to \Bbb{R}$ is an additive function (that is $f(x+y) = f(x)+f(y)$ ...
13
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Continuous versus differentiable

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same ...
13
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5answers
396 views

If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?

Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
13
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A continuous function on $[0,1]$ not of bounded variation

I'm looking for a continuous function $f$ defined on the compact interval $[0,1]$ which is not of bounded variation. I think such function might exist. Any idea? Of course the function $f$ such ...
13
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1answer
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Is there a monotonic function discontinuous over some dense set?

Problem (for fun--not homework) 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 ...
12
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1answer
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Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
12
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1answer
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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 ...
11
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$f(16x)=16f(x) $ and $ f$ is continuous

$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$. Should it be $f(x)=ax$? How can I prove that?
11
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3answers
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If $f(A)\to A^{-1}$, prove that $f$ is continuous.

Let $f \colon GL_{n}(\mathbb{R})\to GL_{n}(\mathbb{R})$ be a function which maps$A\mapsto A^{-1}$. Prove that $f$ is continuous. $GL_{n}(\mathbb{R})=\det^{-1}(\mathbb{R}\setminus\{0\})$ is the ...
11
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A function takes every function value twice - proof it is not continuous

I want to prove the following nice statement I've found: A function $f: [0,1] \rightarrow \mathbb{R}$ takes every function value twice - proof it is not continuous I've already found an answer to my ...
11
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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: $\exists\epsilon>0$ $\forall\delta>0$ ...
11
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3answers
156 views

Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
11
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Homeomorphism between open unit ball and $\mathbb R^n$

Let $B=\{x\in\mathbb R^n : ||x||<1\}$ the open unit ball with the subapce topology of $\mathbb R^n$. I want to show that $B^n\cong\mathbb R^n$ with the map $F(x)=\tan(\frac{\pi ...
11
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Showing continuity of partially defined map

There is a theorem in Note on Cofibrations by Arne Strøm. It says Let $A$ be a closed subspace of a topological space $X$. Then $(X,A)$ has the HEP if and only if there are (i) a neighborhood ...
11
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1answer
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Continuous and Open maps

I was reading through Munkres' Topology and in the section on Continuous Functions, these three statements came up: If a function is continuous, open, and bijective, it is a homeomorphism. If a ...
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Showing that $\Omega$ is of class $C^1$

I have done a lot in this problem, but unfortunately it is not enough to solve it, answers or hints are very welcome. Let $B$ be a rectangle in $\mathbb R^2$ and consider $\varphi\colon ...
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Functions that are continuous only at two points?

I need to find a function $f:\mathbb{R}\to\mathbb{R}$ which is continuous only at two points, but discontinuous everywhere else. How on earth would I go about doing this? I can't think of any ...
10
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does the uniform continuity of $f$ implies uniform continuity of $f^2$ on $\mathbb{R}$?

my question is if $f:\mathbb{R}\rightarrow\mathbb{R}$ is uniformly continuous, does it implies that $f^2$ is so?and in general even or odd power of that function?
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Prove $\sin x$ is uniformly continuous on $\mathbb R$

How do I prove $\sin x$ is uniformly continuous on $\mathbb R$ with delta and epsilon? I proved geometrically that $\sin x<x$ and thus, $$|f(x_1)-f(x_2)|=|\sin x_1 - \sin x_2|\le|\sin x_1|+|\sin ...
10
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1answer
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Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
10
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If $f:\mathbb{R}^n \to \mathbb{R}^n$ is continuous with convex image, and locally 1-1, must it be globally 1-1?

For $f:\mathbb{R}\to \mathbb{R}$ which is continuous, being locally 1-1 implies being globally 1-1, see here. This is not true for a general mapping $f:\mathbb{R}^n\to \mathbb{R}^n$. My intuition as ...
10
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Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
10
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1answer
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Show f is uniformly continuous on $(a,b)$ if it is continuous and $\lim\limits_{x\to a^+}f(x)$ and $\lim\limits_{x\to b^-}f(x)$ exist

Let $f:(a,b)\to\mathbb{R}$ be continuous at all $x\in(a,b)$. If $\lim\limits_{x\to b^-}f(x)$ and $\lim\limits_{x\to a^+}f(x)$ exist in $\mathbb R$, how can we prove that $f$ is uniformly continuous on ...
10
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$g(x)$ is continuous on $\mathbb{R}$ st $g(x)=g(x^2)$. Prove that $g(x)$ is constant.

For $x>0$, $g(\sqrt{x})=g(x)$ and similarly $g(x^{\frac{1}{2^n}})=g(\sqrt{x})=g(x)$ for $n \in \mathbb{N}$ Thus taking $\lim_{n\to \infty}$ both sides we get g(1)=g(x) $\forall x>0$ and ...
10
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1answer
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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 ...