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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13
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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 $x=1$?
20
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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} ...
21
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9answers
790 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
5
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1answer
292 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 ...
3
votes
2answers
190 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 ...
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
votes
2answers
105 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 ...
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1answer
684 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
825 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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1answer
124 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
310 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
349 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 ...
15
votes
6answers
905 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 ...
7
votes
1answer
183 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 ...
10
votes
4answers
6k 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
639 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
114 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 = ...
6
votes
2answers
304 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
1answer
447 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
87 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
138 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
185 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
243 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. $$\lim_{(x,y,x)\rightarrow(0,0,0)}\frac{xyz}{x^2+y^2+z^2}$$
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vote
3answers
715 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 ...
1
vote
1answer
423 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 ...
124
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 ...
28
votes
2answers
669 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
390 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
votes
5answers
385 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
365 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
206 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
351 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 ...
5
votes
2answers
816 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 ...
2
votes
4answers
2k views

limits and Continuity

I had my first encounter with Calculus a decade ago. Back then it was purely mechanical. Formulas and rules of derivation and integration were being written on the board without deriving it and were ...
10
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1answer
1k 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 ...
6
votes
2answers
116 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
198 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 ...
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3answers
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Why Norms are Continuous with details

Please one person describe why norms are continuous function by mathematical symbols.
3
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2answers
211 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 $f(0)<f(1)$ ...
3
votes
1answer
824 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 ...
2
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1answer
62 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
327 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 ...
1
vote
3answers
2k views

Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.

Problem: Let $f$ be defined for all real $x$, and suppose that $$|f(x)-f(y)|\le (x-y)^2$$ for all real $x$ and $y$. Prove $f$ is constant. Source: W. Rudin, Principles of Mathematical Analysis, ...
4
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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
186 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
649 views

Proving that cosine is uniformly continuous

This is what I've already done. Can't think of how to proceed further ...
3
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2answers
468 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
307 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
2answers
93 views

If $f$ has only removable discontinuities, show that $f$ can be adjusted to a continuous function

I was working on this problem from Spivak's text, and I thought I'd post my answer, in case someone can improve on it. In particular, I wonder if there is a proof that can generalize to topological ...