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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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} ...
6
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1answer
503 views

How do I show that all periodic functions are bounded and uniform continuous?

I need help with this question: 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 ...
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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
6
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1answer
327 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
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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
369 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
271 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 ...
4
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4answers
787 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
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2answers
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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 ...
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2answers
708 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 ...
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1answer
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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 $[a, \infty)$.
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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 ...
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 ...
4
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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 ...
10
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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 ...
5
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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.
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3answers
857 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 ...
6
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5answers
523 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
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1answer
137 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
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1answer
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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
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1answer
991 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 ...
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vote
3answers
381 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
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1answer
208 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 ...
7
votes
3answers
128 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 ...
8
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1answer
791 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
157 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
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3answers
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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 = ...
4
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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 ...
2
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1answer
500 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
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1answer
112 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
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2answers
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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
196 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
282 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
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1answer
516 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
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1answer
352 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 ...
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2answers
191 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 ...
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2answers
297 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 ...
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3answers
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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
42 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
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1answer
83 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} ...
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1answer
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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
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3answers
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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 ...
39
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3answers
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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 ...
28
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2answers
789 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
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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
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2answers
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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
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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 ?
8
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1answer
420 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 ...