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 $x=1$?
22
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3answers
1k 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} ...
24
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8answers
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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
5
votes
1answer
320 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 ...
11
votes
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: $\exists\epsilon>0$ $\forall\delta>0$ ...
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 ...
9
votes
2answers
359 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 ...
3
votes
2answers
255 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 ...
3
votes
3answers
678 views

If $f,g$ are uniformly continuous prove $f+g$ is uniformly continuous

Suppose $f:E \rightarrow \mathbb{R}$ and $g:E \rightarrow \mathbb{R}$ are uniformly continuous, where $E$ is a subset of $\mathbb{R}$. Show that $f+g$ is uniformly continuous. What about $fg$ and ...
3
votes
2answers
107 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 ...
1
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2answers
669 views

$f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$

Problem: Suppose $f$ and $g$ are two continuous functions such that $f: X \to Y $ and $g : X \to Y $. $Y$ is a a Hausdorff space. Suppose $f(x) = g(x) $ for all $x \in A \subseteq X $ where ...
-7
votes
1answer
796 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 $[a, \infty)$.
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 ...
12
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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 ...
4
votes
1answer
1k 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 ...
4
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3answers
3k views

Why Norms are Continuous with details

Please one person describe why norms are continuous function by mathematical symbols.
5
votes
5answers
498 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 ...
4
votes
1answer
134 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
1answer
67 views

a continuous mapping is determined by its values on a dense set

Let f and g be continuous mappings of a metric space $X$ into a metric space $Y$ and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$. If $g(p)=f(p)$ for all $p \in E$, prove ...
1
vote
1answer
939 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
3answers
379 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 ...
7
votes
1answer
204 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 ...
6
votes
3answers
126 views

If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex

Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then how do we ...
10
votes
4answers
8k 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 ...
8
votes
1answer
766 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 $f:X \rightarrow Y$ ...
5
votes
1answer
153 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
1k 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 = ...
11
votes
3answers
815 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 ...
2
votes
1answer
2k views

Proof of continuity of Thomae Function at irrationals.

In Thomae Function $$ \begin{align} t(x) = \begin{cases} 0 & \text{if $x$ is irrational}\\ \frac{1}{n} & \text{if $x = \frac{m}{n}$ where $(m,n) = 1$} \end{cases} \end{align} $$ I can prove ...
2
votes
1answer
492 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 ...
1
vote
1answer
110 views

Follow-up regarding right-continuous $f:\mathbb{R} \to\mathbb{R}$ is Borel measurable

I have a follow-up to another question here on math.stackexchange, Are right continuous functions measurable?. The thread was a couple of years old, so I hope it's okay if I start a new question. ...
3
votes
2answers
144 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
195 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
278 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. $$\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$$
2
votes
1answer
504 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 ...
2
votes
1answer
350 views

$|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.

My exercise says: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ a continuous function e suppose that exists $k$ such that: $$|f(x)-f(y)|\geq k|x-y|$$ Then $f$ is bijective and its inverse is ...
1
vote
2answers
182 views

Is every convex function on an open interval continuous?

Let $f:(a,b)\rightarrow \mathbb{R}$. $f$ satisfied the following property: If $\forall x_{1},x_{0},x_{2}\in(a,b)$ and $x_{1}<x_{0}<x_{2};$then$\frac{f(x_{0})-f(x_{1})}{x_{0}-x_{1}}\geq ...
1
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2answers
283 views

For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$

Let $f:X \rightarrow Y$ be a function. Prove that if $f$ is continuous, then for every convergent sequence $(x_n)$ lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ My ...
1
vote
3answers
978 views

Possible to have a continuous sequence?

I'm wondering if it's possible to have a continuous sequence $f: \mathbb{N} \to \mathbb{R}$? My intuition is telling me no because logically it would be impossible to map the natural numbers onto the ...
0
votes
1answer
38 views

Continuous function from a connected set?

If $ f $ is a continuous mapping from a connected set to the real numbers and there exists a real number s that nothing maps to, then the image is either greater than or less than s. this is clear to ...
0
votes
1answer
77 views

About the continuity of $f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} ...
-1
votes
1answer
2k views

Prove the absolute value function of a continuous function is continuous

Suppose that $f$ is a continuous function defined on an interval $I$. Prove that $|f|$ is continuous on $I$. Our definition of continuity: Let $I$ be an interval, let $f:I\rightarrow\Bbb{R}$, and let ...
128
votes
3answers
4k 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 ...
28
votes
2answers
765 views

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 ...
11
votes
3answers
400 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 ...
6
votes
2answers
2k views

Dini's Theorem. Uniform convergence and Bolzano Weierstrass.

In Spivak's chapter on uniform convergence he asks to prove the following THEOREM Let $\{f_n\}$ be sequence of continuous functions that converge pointwise to $0$ over $[a,b]$. If $0\leq ...
13
votes
5answers
392 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
416 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
214 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 ...
9
votes
7answers
580 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 ...