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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17
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How to show that $f(x)=x^2$ is continuous at $x=1$? [closed]

How to show that $f(x)=x^2$ is continuous at $x=1$?
22
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3answers
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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} ...
10
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1answer
2k views

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 ...
25
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7answers
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A map is continuous if and only if for every set, the image of closure is contained in the closure of image

As a part of self study, I am trying to prove the following statement: Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if ...
17
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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
495 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 ...
29
votes
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
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2answers
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Proof of continuity of Thomae Function at irrationals.

In Thomae's Function: $$ \begin{align} t(x) = \begin{cases} 0 & \text{if $x$ is irrational}\\ \frac{1}{n} & \text{if $x = \frac{m}{n}$ where $\gcd(m,n) = 1$} \end{cases} \end{align} $$ I ...
7
votes
5answers
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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 ...
46
votes
3answers
3k views

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 ...
29
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5answers
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What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference geometrically? What is the best way to describe the difference between these two concepts? Where the motivation of ...
-1
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1answer
536 views

Every uniformly continuous real function has at most linear growth at infinity

Assuming $f:\mathbb R\to\mathbb R $ be an uniform continuous function, how to prove $$\exists a,b\in \mathbb R^+ \quad \text{such that}\quad |f(x)|\le a|x|+b.$$
6
votes
1answer
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Continuity of a convex function

I'm trying to solve the following problem: Let $f:K\rightarrow \mathbb{R} $, $f$ convex and $K \subseteq \mathbb{R}^n$ convex. Then $f$ is continuous on $K$. I have proved the only case $n=1$, but ...
5
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1answer
4k views

Is an integral always continuous?

Say I have a function f(x) on some interval [a,b]. Say it is integrable such that $\int f(x)~dx $ is defined. Is $\int f(x)~dx $ necessarily continuous? If I were to know that the integral is ...
5
votes
2answers
426 views

Ideal in a ring of continuous functions

Let $R$ be the ring of all continuous real valued functions on the unit interval $[0,1]$ (with pointwise operations), and let $I$ be a proper ideal of $R$. Show that there exists $λ\in [0,1]$ such ...
17
votes
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 ...
8
votes
1answer
399 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
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 ...
9
votes
1answer
1k views

Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$. Is $f$ continuous? Let $f$ be ...
2
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1answer
219 views

$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.

$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$. I know f uniformly continuous $\implies \forall\varepsilon > 0s.t.\forall ...
15
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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 ...
11
votes
2answers
506 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 ...
7
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3answers
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Why are norms continuous?

Describe why norms are continuous function by mathematical symbols.
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2answers
193 views

Continuous function positive at a point is positive in a neighborhood of that point

Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) > 0$ then there exists a $d > 0$ such that $\forall x$, $f(x) > 0$ with $|x-c| < d$. I can understand ...
4
votes
1answer
569 views

Continuity of absolute value of a function

Let $f(x)$ be a continuous function. Prove that $\left|f(x)\right|$ is also continuous. Is it correct to say that, by the reverse triangle inequality, $\left|f(x)-f(c)\right| \geq ...
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2answers
1k views

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

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 ...
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2answers
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$f,g$ 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 ...
4
votes
4answers
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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 ...
4
votes
3answers
784 views

sequentially continuous on a non first-countable

Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this ...
3
votes
2answers
115 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 ...
26
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1answer
530 views

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, ...
12
votes
1answer
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Equicontinuity on a compact metric space turns pointwise to uniform convergence

I know that If $\{f_n\}$ is an equicontinuous sequence, defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly. I'm having trouble ...
14
votes
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 ...
5
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1answer
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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 ...
9
votes
7answers
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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 ...
11
votes
3answers
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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 ...
6
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1answer
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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 ...
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votes
1answer
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Prove the absolute value function of a continuous function is continuous [duplicate]

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 ...
6
votes
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 ...
5
votes
1answer
461 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 ...
5
votes
3answers
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Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?

Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies $$\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0.$$ Does this imply that $f$ is continuous? Source: W. Rudin, Principles of ...
4
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1answer
148 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 ...
2
votes
2answers
1k 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 ...
2
votes
1answer
165 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
46 views

Increasing real valued function whose image set is connected

Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true: a) $f$ is discontinuous exactly at one point. b) $f$ is ...
1
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1answer
77 views

Prove symmetry of probabilities given random variables are iid and have continuous cdf

Let $Y_1, Y_2, ...$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_{Y}(y) := F_{Y_1}(y) = F_{Y_2}(y) ...
1
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1answer
2k 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
2answers
229 views

Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ is a continuous function using epsilon-delta.

THE QUESTION: Use the metric $(x,y)$ = $\rho(x,y)=|x-y|$ for the reals and use the metric $\rho((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ for the plane. Define $f:R\times R \to R$ as ...
1
vote
3answers
467 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 ...
0
votes
3answers
463 views

If a function is undefined at a point, is it also discontinuous at that point?

I posted a solution here with an illustration (see below) and commented that the function was discontinuous at $x=1$, where it is undefined. Someone told me, no, it is undefined but continuous. Now ...