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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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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3answers
909 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} ...
20
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9answers
651 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
10
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3answers
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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: $\exists\epsilon>0$ $\forall\delta>0$ ...
3
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2answers
98 views

Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $

Let $f$ a continous function defined in the interval $[0,1]$. Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $ I tried to use Heine–Cantor theorem ...
-3
votes
1answer
576 views

Prove uniform continuity of a continuous function

Suppose $f$ is continuous on $[a, \infty)$ and that the limit (as $x$ approaches infinity) is $L$ for some real number $L$. Prove that $f$ is uniformly continuous on $[a, \infty)$.
4
votes
1answer
741 views

Is the total variation function uniform continuous or continuous?

I have been doing some excercises on total variation when the following questions came up to my mind: (1) Let $f$ be continuous on the interval $[0,1]$ and be of bounded variation. Is it true that ...
3
votes
1answer
254 views

Continuous map $f : \mathbb{R}^2\rightarrow \mathbb{R}$

Let $f : \mathbb{R}^2\rightarrow \mathbb{R}$ be a continuous map such that $f(x)=0$ only for finitely many values of $x$. Which of the following is true? Either $f(x)\leq 0$ for all $x$ or ...
4
votes
1answer
120 views

A problem on continuous functions

$f : S^1 \rightarrow \mathbb{R}$ is a continuous map. Define $$A = \{(x, y) \in S^1 \times S^1: x \neq y, f(x) = f(y)\}$$ We want to prove that A has uncountably many points. It seems very evident, ...
3
votes
4answers
218 views

Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?

Let $f\colon[0,1]\to\mathbb{R}$ be continuous such that $f(x)\in\mathbb{Q}$ for any $x\in[0,1]$. Intuitively I feel that $f$ is constant, since $\mathbb{Q}$ is dense in $\mathbb{R}$. How can I ...
1
vote
3answers
328 views

Fixed point in a continuous function

Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that ...
14
votes
6answers
739 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 ...
5
votes
1answer
161 views

Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
8
votes
1answer
535 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 $f:X \rightarrow Y$ ...
4
votes
1answer
81 views

Continuous function on closed unit ball

Take a continuous mapping $f: \bar{B^{n}} \rightarrow \bar{B^{n}}$, where $\bar{B^{n}}$ is a closed unit ball in $\mathbb{R}^{n}$. Assume that $f(x) \neq x$ for every $x \in \bar{B^{n}}$. Define ...
2
votes
3answers
784 views

Uniform Continuity of $f(x)=x^3$

1.)Determine whether $f(x)=x^3$ is uniformly continuous on [0,2) So far, I have $\delta$ = 2 and $\epsilon$ = 8, and plan on using the sandwich theorem with $x^2$ and eventually equating $\delta = ...
9
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4answers
5k views

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 ...
6
votes
2answers
277 views

$f$ brings convergent nets to convergent nets, is it continuous?

Let $f:(X,\mathcal T)\to (Y,\mathcal S)$ be a function between topological spaces. Let for any convergent net $(x_\alpha)$ in $X$, $(f(x_\alpha ))$ be convergent in $Y$. Is $f$ continuous? (It seems ...
2
votes
2answers
147 views

Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
2
votes
1answer
408 views

What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$? [duplicate]

Possible Duplicate: Is there a name for such kind of function? I am investigating functions satisfying the exponentiation identity $f(a + b) = f(a)f(b)$ for all $a,b\in \Bbb R$. This is ...
3
votes
2answers
135 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
2
votes
3answers
182 views

Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$

Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$ Going from the epsilon delta definition we get: $$\forall x,y>1,\text{WLOG}:x>y \ ,\ \forall\epsilon>0,\exists\delta>0 ...
2
votes
4answers
217 views

Find the limit (if it happens to exist or prove it doesnt exist)

Find the limit (if it even exists). If not, prove it doesn't exist. $$\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$$
1
vote
1answer
347 views

proof of that a continuous function has a fixed point

Can you please help me to understand this proof: Consider $g(x)=f(x)-x$. $f(a)\ge a$ so $g(a)=f(a)-a\ge 0$. $f(b)\le b$ so $g(b)=f(b)-b\le 0$. By the Intermediate Value Theorem, since $g$ is ...
119
votes
3answers
3k views

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 ...
11
votes
3answers
379 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 ...
14
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5answers
367 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 ?
8
votes
1answer
292 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 ...
4
votes
1answer
200 views

Limit with integral or is this function continuous?

Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center ...
8
votes
7answers
252 views

How to prove continuity of $e^x$.

I simply want a proof that $e^x$ is continuous. I have never really been able to find something satisfying these points: $e$ is defined to be the limit $\lim_{n\to\infty}\left(1+{1\over ...
4
votes
2answers
379 views

Does there exist an unbounded function that is uniformly continuous?

I know that $1/x$ is unbounded on $(0,5)$ (for example) and that since it is unbounded, it is not uniformly continuous. Does a function have to be bounded to be uniformly continuous? I don't think ...
6
votes
2answers
81 views

Tough Continuous function Limit Question

Suppose f(x) and g(x) are continuous functions on [a,b] with f monotone increasing. Assume there exists a sequence $x_n \in [a, b]$ such that for all $n \in N$ , $g(x_n) = f(x_{n+1})$. Show that there ...
6
votes
1answer
176 views

Existence of a power series converging non-uniformly to a continuous function

I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that: $f$ converges and is continuous on the closed unit disk $D$ and the series $\sum_n a_n z^n$ does not converge ...
3
votes
2answers
181 views

Continuous mapping $f: [0,1]\rightarrow (0,1)$ CSIR December $2013$

Question is : Suppose $f: [0,1]\rightarrow (0,1)$ is Continuous then which of the following is NOT true.. $F\subseteq[0,1]$ is closed set implies $f(F)$ is closed in $\mathbb{R}$ If $f(0)<f(1)$ ...
2
votes
1answer
35 views

A continuous function that attains neither its minimum nor its maximum at any open interval is monotone

Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq \mathbb R$ , then how to prove that $f$ is ...
2
votes
4answers
314 views

Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, … $ but is continuous at every other point

(a) Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, ... $ but is continuous at every other point. (b) Find a function $f:\Bbb R \to \Bbb R$ which is ...
8
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1answer
878 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 ...
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3answers
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Proof for Dirichlet Function and discontinuous

I think I don't understand how it works.. I found some proofs.. okay, let's see: Well I'd like to show that the function, $$f(x) = \begin{cases} 0 & x \not\in \mathbb{Q}\\ 1 & x \in ...
4
votes
1answer
170 views

prove $x \mapsto x^2$ is continuous

I am to show the continuity of this function with the help of $\epsilon$-$\delta$ argument. The function is: $g: \Bbb{R} \rightarrow \Bbb{R}$, $x \mapsto x^2$. Given the $\epsilon$-$\delta$ ...
4
votes
2answers
479 views

Proving that cosine is uniformly continuous

This is what I've already done. Can't think of how to proceed further ...
3
votes
1answer
704 views

Show that $f$ is uniformly continuous if limit exists

Let $f(x)$ be continuous on $(0,1]$. Show that $f$ is uniformly continuous IFF $\displaystyle \lim_{x\to0^+} f(x)$ exists. Thoughts: Backward Proof: Let another function $\overline f(x)$ be ...
3
votes
2answers
431 views

prove that $f(x)=\log(1+x^2)$ is Uniform continuous with $\epsilon ,\delta$ …

I have to prove that $f(x)=\log(1+x^2)$ is Uniform continuous in $[0,\infty)$ (with $\epsilon ,\delta$ formulas...) I wrote the definition: (what I have to prove): $\forall \epsilon>0 \quad ...
3
votes
2answers
271 views

Continuous mappings pull back closed sets to closed sets

George F Simmons, Topology and Modern Analysis pg.79 Problem 4 Let $X$ and $Y$ be metric spaces. Show that an into mapping $f:X \rightarrow Y$ is continuous $\iff$ $f^{-1}\left(G\right)$ is closed in ...
2
votes
1answer
92 views

When is $x^{\alpha}\sin(x^{\beta})$ uniformly continuous for $\alpha, \beta > 0$?

Consider a function $f_{\alpha, \beta}\colon (0, \infty) \longrightarrow \mathbb{R}$ defined in the following way: $$f_{\alpha, \beta} = x^{\alpha}\sin(x^{\beta}) \quad \alpha, \beta > 0$$ Then ...
2
votes
1answer
75 views

Proving and disproving $\exists b,c\in \mathbb R$ such that $f(x)=\frac a2x^2+bx+c $

Let $f:I\to\mathbb R$ where $I$ is an interval, $f''(x)=a \ \ \forall x\in I$. Prove that there exsits such numbers $b,c\in \mathbb R$ such that: $f(x)=\frac a2x^2+bx+c ,\ \forall x\in I$. ...
2
votes
1answer
61 views

Proofs involving sequential limit

Let $S$ be the domain of the function $f$. Suppose $S=\left\{\frac{1}{n}: n\in\Bbb N\right\}$. Show $\lim_{x\to0}f(x)=L$ iff $\lim_{n\to\infty}f\left(\frac{1}{n}\right)=L$. Idea: I want to say, let ...
2
votes
1answer
171 views

Why are the Differential- and multiplication mapping on $C^{\infty}(\Omega)$ continuous?

Let $\Omega\subset\mathbb{R}^n$ be open and $\Omega\neq\varnothing$ and suppose we have the Fréchet topology on $C^{\infty}(\Omega)$ (this can be obtained by the topology construction from out ...
1
vote
1answer
124 views

Prove continuity for cubic root using epsilon-delta

I am trying to prove that a function is continuous at a point a using the $\epsilon$-$\delta$ theorem. I managed to find a $\delta$ in this case $|2x^2+1 - (2a^2+1)| < \epsilon$. But I have a hard ...
1
vote
1answer
66 views

Find a uniformly continuous function such that $a_{n+1}=f(a_n)$

$a_{n+1} = a_n - a_n^2$, $a_1 = 2/3$. for $n\ge1$ a) Show the series converges and find its limit. b) find a uniformly continuous $f:\mathbb{R}\rightarrow \mathbb{R}$ such that: ...
1
vote
1answer
113 views

$f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...