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) ...

learn more… | top users | synonyms (1)

1
vote
4answers
65 views

Why isn't the initial topology always the trivial topology?

If I have a set $X$ and a function $f:X\rightarrow X$, then I think $f$ is continuous with the trivial topology, because no matter what the function is, $f(X)\subseteq X$. Thus for any point $f(x)$, ...
2
votes
2answers
35 views

Limit of a function with 2 variables

I am given this function: $$f(x,y)=\begin{cases}\frac{xy^3}{x^2+y^4} & \text{ for } (x,y)\not=(0,0)\\ 0 & \text{ for } (x,y)=(0,0)\end{cases}$$ and I have to check if it is continuous in ...
1
vote
1answer
23 views

Convergence in probability of a composite function.

Question: Let $f$ be a continuous function on $\mathbb{R}.$ If $X_n \to X$ in probability, then $f(X_n) \to f(X)$ in probability. The result is false if $f$ is merely Borel measurable. [Hint: ...
3
votes
2answers
54 views

Proving characterization of continuity with direct images of sets using nets.

We know that if $X,Y$ are topological spaces, then $f: X \to Y$ is continuous if and only if $f(\overline{E}) \subseteq \overline{f(E)}$, for all $E \subseteq X$. I started studying nets by myself ...
0
votes
5answers
269 views

Is sine of one degree a real? If not, how is sine continuous?

If I understand it correctly, the impossibility of trisection of an arbitrary angle implies that sine of one degree isn't a real number, but how is it then possible for sine to be continuous, if it ...
3
votes
2answers
35 views

Proving that $(X,\tau)$ is Hausdorff given a condition.

Let $(X,\tau)$ be a topological space such that for each $p \in X$ there is a continuous function $f:X \to \Bbb R$ verifying $f^{-1}(\{0\}) = \{p\}$. Then $(X,\tau)$ is Hausdorff. Welp, take $p,q ...
1
vote
2answers
42 views

What kinds of functions have fixed points?

Among continuous functions, can we characterize those which have fixed points and those which do not? Geometrically, these are the functions that intersect the line $f(x) = x$. Is that the most ...
3
votes
1answer
105 views

Computing the volume of this weird object,

Let $f: [-1,1] \to \mathbb{R}$ be a continuously differentiable function such that $f(-1) = f(1) = 0$ and $0<f(x)\le 1$ for all $x \in (-1,1)$. Let $S$ be the surface in $\mathbb{R}^3$ obtained by ...
1
vote
2answers
45 views

Condition for function $f(x)=\frac{(1-x)^{-1/2}-(1+x)^{1/2}}{(1-\frac{x}{2})^{-1/2}-(1+\frac{x}{2})^{1/2}},(x\neq0)$ to be continuous at $x=0$.

This function is not continuous at $x=0$. I know that function (in the example) is continuous if $$\lim\limits_{x\to0^-}f(x)=\lim\limits_{x\to0^+}f(x)=f(0)$$ and limits and $f(x_0)$ must be defined. I ...
-3
votes
1answer
37 views

Coninutity of this function in interval $(0,1)$ [closed]

Let $f(x)$ be the function defined on the interval $(0,1)$ by $$ f(x) = \begin{cases} x(1-x) \quad\text{if}\quad x \in \Bbb Q \\ \frac{1}{4}-x(1-x) \quad\text{if}\quad x \in ...
0
votes
2answers
33 views

Continuity of increasing function [duplicate]

If $f$ is an increasing function over the reals, given a number $M$, is it always possible to find some $x \ge M$ such that $f$ is continuous at $x$? This seems like it should be intuitively true but ...
1
vote
3answers
62 views

Show that the function $(x^2+y^4)f(x,y)+f(x,y)^3=1$ is $C ^ 1$ class.

Consider $f:U\subset\mathbb{R}^2 \rightarrow \mathbb{R}$ a continuous function in open set $U$. Show that $(x^2+y^4)f(x,y)+f(x,y)^3=1$ is $C ^ 1$ class, for all $(x,y) \in U$. I think we can use the ...
2
votes
0answers
42 views

What are the restrictions on using substitution in integration?

* One photo is equal 1000 words. * Integration done by substitution $u=\tan x$. Integration done by substitution $u=\tan {x\over 2}$. The source function is a continuous positive function ...
2
votes
3answers
62 views

Is there any function $f:\mathbb R \rightarrow \mathbb R$ such that it is only continuous at rational numbers?

Is there any function $$f:\mathbb R \rightarrow \mathbb R$$ such that it is only continuous at rational numbers?
1
vote
1answer
86 views

$f:\mathbb R\to \mathbb R$ continuous, $f(f(0))=0$ so there exists $a \in \mathbb R$ such that $f(2a)=3a$

Let $f:\mathbb R\to \mathbb R$ continuous such that $f(f(0))=0$. Prove that there exists $a \in \mathbb R$ such that $f(2a)=3a$. Well, I figured that in such exercises, I should define a new function ...
0
votes
2answers
25 views

If $f$ is continuous and piecewise $C^1$ and $f'$ is bounded a.e., is $f$ Lipschitz?

If $f$ is continuous and piecewise $C^1$ on $\mathbb{R}$ (only a finite number of pieces) and $f'$ is bounded a.e., is $f$ globally Lipschitz? So $f$ is only not differentiable in a finite number of ...
-1
votes
2answers
41 views

continuous and monotonic function [duplicate]

If there is a function that continuous in a interval monotonic in the same interval Does it mean the function is also differentiation function in the interval?
0
votes
2answers
30 views

continuity of inverse function

I studied derivative of function ${f^{-1}}'(y)=\frac1{{f}'(x)}$ When I tried above proof , it needs continuity of inverse function At this point , I have a question $f$ is continuous on D , then ...
0
votes
0answers
28 views

Is this multivariable function continuous?

My function is: $$f\left(x,\:y\right)\:=\:y\left(sin\left(\frac{1}{x-1}\right)\right)\::\:x\:\ne 1$$ $$f\left(x,\:y\right)\:=\:0\::\:x\:=1$$ The question sounds like: "Are this functions continuous?" ...
1
vote
0answers
19 views

Is the continuous extension theorem true when the range space of $f$ is not complete?

So the problem is Exercise $13$, Chap. $4$ of Principles of Mathematical Analysis by Rudin: Problem Let $E$ be a dense subset of metric space $X$, and let $f$ be a uniformly continuous real function ...
1
vote
2answers
31 views

Continuous of function in a point

Given the function $$f(x,y)=\frac{1-\cos(2xy)}{x^2y^2}$$ I want the function to be continuous in $(0,0)$. If I assume that the limit when $x\rightarrow0$ equals to the limit when $y\rightarrow0$, I ...
2
votes
3answers
99 views

Continuous bijective function between the same topology that is not a homeomorphism.

I know there are many examples when the domain and co-domain do not coincide. Taking the identity on $X$ from $(X,\tau_1)$ to $(X,\tau_2)$ when $\tau_2$ is coarser than $\tau_1$ gives an infinite ...
1
vote
1answer
30 views

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ ...
3
votes
2answers
60 views

Prove that $U-f(U)$ is an open set.

Let $(X,d)$ be a compact metric space. Let $f:X\to X$ be continuous. Fix a point $x_0\in X$, and assume that $d(f(x),x_0)\geq 1$ whenever $x\in X$ is such that $d(x,x_0)=1$. Prove that $U\setminus ...
1
vote
1answer
44 views

show that the function $\{x_n\}\mapsto \sum_{n=1}^\infty 2^{-n}x_n$ is continuous

This problem comes from an old Preliminary exam: Consider the space $[0,1]\times [0,1]\times \cdots$ (the countably infinite product of $[0,1]$ with the product topology) An element of $X$ may be ...
0
votes
0answers
38 views

Trying to prove that if $f:[a, b]\to[s, t]$ is monotone then $f$ is continuous

I'm trying to prove that if $f:[a, b]\to[s, t]$ is monotone (and its image is closed interval) then $f$ is continuous. My attempt: I say wlog, $f$ is increasing. I know that a monotone function only ...
1
vote
1answer
110 views

IMC 2008 first problem first day. Finding continuous functions so $x-y\in \mathbb Q \implies f(x)-f(y)\in \mathbb Q$

I would like an alternate solution and proof verification for the following problem: Find all continuous functions $f:\mathbb R \rightarrow \mathbb R$ so that if $x-y$ is rational then $f(x)-f(y)$ is ...
3
votes
1answer
37 views

How can a boundary measure of a function be absolutely continuous?

I'm studying firsts tools in several complex variables. In my book I found what follows: It can be proved that if $\varphi$ is strongly subharmonic and has a finite majorant in the unit ball, then ...
9
votes
0answers
98 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
3
votes
1answer
68 views

How do I show $\lim_{x\to\infty}f(x) = \lim_{x\to\infty} f '(x)=0$ if $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0$? [duplicate]

$f(x)$ is a real valued function on the reals, and has a continuous derivative such that $$\lim_{x\to\infty} f'(x)^2 + f(x)^3 = 0.$$ How do i show that $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} ...
0
votes
1answer
20 views

Determining continuity and differentiability

Is this function continuous and differentiable? $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x\:\ge 1 \end{array}\right.$$ For continuity, I did $$\lim_{x\to 1^+\:}f(x) = ...
0
votes
2answers
39 views

Analysis of continuity and differentiability of a function

Find a,b,c $\in \mathbb{R}$ for which the function is a) continuous, b) differentiable. $$f(x)=\left\{\begin{array}{cc} ax^2+bx+c & x<0 \\ 2\sin x+cos x & x\:\ge 0 \end{array}\right.$$ ...
1
vote
0answers
24 views

$f$ is monotone on D and $f(D)$ is an interval

$f$ is monotone on D and $f(D)$ is an interval then $f$ is continuous Is my proof right? pf) First, suppose it is monotone increasing Since $f(D)$ is an interval there is $[c,d]$ such that ...
1
vote
1answer
80 views

The set of continuity of a pointwise limit of continuous functions

Let $\{x_n(t)\}_{n=1}^{\infty}$ be real a sequence of continuous function from $[0,1]$ to $\mathbb{R}$, and $\{x_n(t)\}_{n=1}^{\infty}$ converges pointwise to $x(t)$ i.e. $\lim_{n \to \infty} x_n(t) ...
0
votes
2answers
36 views

Is it true that a mapping between metric spaces is continuous iff the image of every open set is open?

Just want to change Rudin theorem 4.8 a bit and see if this works. The original theorem is ... $f$ is continuous iff $f^{-1}(V) $ is open in $X$ for every open set $V$ in $Y$. If I change the ...
2
votes
3answers
49 views

Continuity of function consisting of an infinite series.

Let $f(x) , 0\leq x\leq 1$ be defined by, $$f(x)=\sum_{n=1}^{\infty}\frac{1}{(x+n)^2}$$. Show that $f$ is continuous on $[0,1]$ and that, $$\int_0^1f(x)dx=1$$. I have never dealt ...
3
votes
1answer
429 views

Is the Sinc function continuous?

Is $\frac{\sin x}{x}$ a continuous function or is it not? I am confused with the fact that at zero it cannot be defined yet the limit surely exists. So, the question of its continuity arises.
0
votes
2answers
56 views

Continuity in $\mathbb R$ results in continuity in $\mathbb R^2$; Proof?

During studying of proof of some other theorem, I faced with the claim (without proof): since $f(x,t)$ and $g(x,t)$ are continuous functions [$f,g:\mathbb R^2 \rightarrow \mathbb R$] thus the ...
0
votes
1answer
14 views

Upper semi-continuity results

I have recently been introduced to the notion of upper semi-continuity on a metric space $X$. Please advise on the following queries: If $f:X \rightarrow \mathbb{R}$ is upper semi-continuous and ...
0
votes
2answers
67 views

Is my proof for this limit correct?

I want to prove that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to 2. Let $a_0$ = $\sqrt{2}$ $a_n$= $\sqrt{2+a_{n-1}}$. Then, proving that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to ...
0
votes
1answer
11 views

Limiting and continuous about one function

I have a function which is \begin{equation} F(x)= \begin{cases} f(x) & x \in [\underline{x},\bar{x})\\ \\ f(\bar{x}) & x=\bar{x} \end{cases} \end{equation} The function $f(x)$ is strictly ...
0
votes
1answer
31 views

which hypothesis for boundedness of this function

Let $v:[0,\infty)\rightarrow \mathbb{R}_+$ be a positive function such that $$\exists T,q>0\,\,s.t.\,\, \forall t\in[0,\infty),\,\,\int_t^{t+T} v(\tau) d\tau \le q$$ I'm looking for the "less ...
2
votes
1answer
51 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
-2
votes
1answer
34 views

Positive derivative on [0,1] implies a continuous derivative on [0,1]

If a real-valued function F defined on [0,1] is differentiable with positive derivative f everywhere on [0,1], can we conclude that f is continuous?
1
vote
1answer
61 views

Alternative Proof of the Extreme Value Theorem

I have proven the Boundedness Theorem for continuous functions and would now like to prove the Extreme Value Theorem; that is, show that the upper bound is indeed attained for continuous functions. I ...
0
votes
1answer
28 views

Distance of a point to a subset.

Let $(M,d)$ be a metric space. For a subset $A\subseteq M$ we define the distance of a point $x$ to $A$ as $$\alpha_A(x):=\operatorname{dist}(x,A):=\inf_{y\in A}d(x,y)$$ Prove that: ...
1
vote
1answer
29 views

Continuity of composite functions

The continuity theorem for composite functions states that if $f(x)$ is continuous at $x = a$ and $g(x)$ is continuous at $x = a$ , then the composite function $f\circ g$ and $g\circ f$ are also ...
3
votes
0answers
60 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
0
votes
2answers
19 views

Is homeomorphic image of closed bounded subsets of metric spaces , also closed bounded in the homeomorphic image metric space?

Let $X$ , $Y$ be homeomorphic metric spaces with homeomorphism $f$ , then is it true that for any closed bounded subset $A$ of $X$ , $f(A)$ is also closed and bounded in $Y$ ?
0
votes
0answers
25 views

If for every $a > 0$, $u \in C^\infty([a,\infty))$, then is $u \in C^\infty((0,\infty))$?

Suppose that for every $a > 0$, $u \in C^\infty([a,\infty))$. Does this imply that $u \in C^\infty((0,\infty))$? I think it is true when we just work in $C^0$, but with $C^\infty$ you need to ...