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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118
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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 ...
29
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9answers
733 views

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 ...
19
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3answers
872 views

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} ...
18
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1answer
253 views

$f(f(\sqrt{2}))=\sqrt{2}$ 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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2answers
477 views

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 ...
17
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6answers
785 views

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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9answers
474 views

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
16
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1answer
297 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 ...
15
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2answers
491 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to 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 ...
14
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6answers
618 views

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 ...
14
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5answers
352 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 ?
14
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2answers
727 views

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 ...
14
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1answer
316 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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2answers
169 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 ...
11
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8answers
5k views

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 ...
11
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4answers
363 views

$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
395 views

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 ...
11
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1answer
708 views

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 ...
11
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1answer
268 views

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 ...
10
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3answers
175 views

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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3answers
378 views

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 ...
10
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1answer
366 views

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 ...
10
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3answers
369 views

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 ...
10
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1answer
737 views

$f(x)=x$ if $x$ irrational and $f(x)=p\sin\frac1q$ if $x$ rational

Define the real-valued function $f$ on $\mathbb{R}$ by setting $f(x)=x$ if $x$ is irrational, and $f(x)=p\sin\frac1q$ if $x=\frac{p}q$ is written in lowest terms. At what points is $f$ continuous? ...
9
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4answers
385 views

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?
9
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5answers
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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 $x=1$?
9
votes
1answer
175 views

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 ...
9
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1answer
315 views

Proving the existence of a point with a certain property for a continuous function

Let $f:[0,1]\to\mathbb{R}$ a continuous function and $\int_0^1xf(x)dx=0$. Show that there exists a point $c\in(0,1)$ so that $f(c)=(\int_c^1f(x)dx)^2$. As a potential solution, I tried assuming that ...
9
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3answers
289 views

Solve the functional equation $f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)$ with $f : [0,\infty) \to \mathbb R$ continuous

Solve the functional equation $$f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)\qquad \forall x\geq 0$$ with $f : [0,\infty) \to \mathbb R$ continuous. I can't manage to get this one ...
9
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1answer
108 views

Continuous functions with a certain condition

What is the class of continuous functions $f\colon \mathbb{R}\to\mathbb{R}$ which satisfy $f(x)-f(y)\in\mathbb{Q}$ if and only if $x-y\in \mathbb{Q}$?
9
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1answer
498 views

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 ...
8
votes
2answers
735 views

Give an example of a function $h$ that is discontinuous at every point of $[0,1]$, but with $|h|$ continuous on $[0,1]$

Give an example of a function $h:[0,1]\to\mathbb{R}$ that is discontinuous at every point of $[0,1]$, but such that the function $| h |$ that is continuous on $[0,1]$. I don't really even know where ...
8
votes
5answers
441 views

Study continuity of this function

Hello im studying calculus at the university and I dont know how to solve the following exercise: Study the continuity of the next function: $$f(x,y) = \begin{cases} \frac{x^2-xy}{x+y}&\text{for } ...
8
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5answers
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What are the points of discontinuity of $\tan x$?

$f(x) = \tan x$ is defined from $\mathbb R - \{\frac{\pi}{2} (2n+1) \mid n \in \mathbb Z\}$ to $\mathbb R$. For every $x$ in its domain, $$f(x) = \frac{\sin x}{\cos x}$$ where $\cos x$ is never 0. ...
8
votes
3answers
433 views

Are there any geometric interpretations to uniform continuity?

There are specifically the two forms, continuity and uniform continuity, I'm referring to. So a function is continuous if the graph "doesn't break," but this also applies to a uniform continuous ...
8
votes
3answers
242 views

Does this intuition for “calculus-ish” continuity generalize to topological continuity?

In the past, I've always motivated continuity of a function from (some subset of) $\mathbb R$ to $\mathbb R$ based on the (incomplete) definition $\lim_{x \to c} f(x) = f(c)$; continuity at isolated ...
8
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4answers
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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 ...
8
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1answer
492 views

Is a continuous function simply a connected function?

Intuitively, a function $\mathbb{R}\rightarrow\mathbb{R}$ is continuous if you can draw its graph without taking the pen off the page. This suggests the following theorem: A map $f:X \rightarrow Y$ ...
8
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1answer
765 views

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 ...
8
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1answer
64 views

$f^{-1}(U)$ is regular open set in $X$ for regular open set $U$ in $Y$, whenever $f$ is continuous.

Let $f$ be a continuous function from space $X$ to space $Y$. If $U$ is regular open set in $Y$, it it true that $f^{-1}(U)$ is a regular open set in $X$?
8
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1answer
280 views

pointwise limit on a complete metric space

Let $\{f_n: X\rightarrow \mathbb{R}\}$ be a sequence of continuous real-valued functions on a complete metric space, $X$. Suppose this sequence has a pointwise limit, $f$. How easy is it to see that ...
8
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3answers
2k views

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: $\exists\epsilon>0$ $\forall\delta>0$ ...
8
votes
3answers
193 views

Continuous extension of analytic functions

Is it possible to prove the following statement or is there a counter-example: Let $H=\{y>0\}$ be the upper half plane in the complex plane. If $f$ is an analytic function on $H$ and its real part ...
8
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1answer
154 views

Path Connectedness and continuous bijections

Mathoverflow. Are there any two topological spaces $X$ and $Y$ such that they are path connected and such that there exist continuous bijections $X\rightarrow Y$ and $Y\rightarrow X$, but and yet ...
7
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9answers
341 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
7
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3answers
583 views

I need to prove that $f$ continuous at $(x)=0$ using a $\epsilon$-$\delta$ proof

I need to prove that $f$ continuous at $(x)=0$ using a $\epsilon$-$\delta$ proof $$ f(x, y) = \begin{cases} x^2sin(\frac1x),&x\neq 0 \\ 0,&x = 0 \end{cases} $$
7
votes
1answer
148 views

How can a $C^1$-continuous surface have infinite curvature?

Short version: Apparently it is possible for a $C^1$-continuous parametric surface to (locally) have infinite (Gaussian) curvature. I find this quite counter-intuitive, because I always thought that ...
7
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1answer
149 views

Continuous surjective functions $\omega_1 \to \omega_1$.

I am looking for nontrivial examples of surjective continuous functions from $\omega_1$ onto $\omega_1$ (with both $\omega_1$'s in the order topology). What sorts of properties must these functions ...
7
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2answers
136 views

Continuous functions on $\mathbb{R}^2$ with special property

The following problem is from Miklos Schweitzer competition (Year 1983, Problem 7): Prove that if the function $f: \mathbb{R}^{2}\to [0, 1]$ is continuous, and its average on every circle of ...
7
votes
1answer
239 views

Dini's continuity vs Holder continuity

(listed items are just the definitions, you can skip to "Clearly" if you are familiar with them) Let $E \subset \mathbb{R}^N$ and let $f \colon E \to \mathbb{R}.$ The modulus of continuity of $f$ is ...