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
1answer
52 views

Rudin 4.22 Let $f$ be a continuous function from $X$ to $Y$ (metric spaces). If $E$ is connected, then $f(E)$ is connected

Rudin 4.22. Let $f$ be a continuous function from $X$ to $Y$ (metric spaces). If $E$ is connected, then $f(E)$ is connected. Could someone check this proof: Proof: I will show the ...
1
vote
0answers
16 views

Is this function strongly convex?

Let A,B be two intervals in $R$ and let $f(x,y):A\times B\rightarrow R $ be a continues function. Assume that $f$ is convex in both $x$ and $y$ Is the following function $g:A \rightarrow R$ is ...
2
votes
2answers
48 views

If $f:\mathbb R^n\to\mathbb R$ is twice continuously differentiable, then $\nabla f$ is Lipschitz continuous

Let $f\in C^2(\mathbb R^n)$. I've read that since $f$ is twice continuously differentiable, $\nabla f$ is Lipschitz continuous. Is that really true? By the mean-value theorem, $$\left\|\nabla ...
2
votes
1answer
45 views

A Map From $S^n\to D^n/\sim$ is Continuous.

The following question is motivated by Thomas's answer here which can be used to prove that $\mathbf RP^n$ is same as the space obtained by identifying the antipodal points on the boundary circle of ...
5
votes
1answer
36 views

Characterizing the continuous functions from $\mathbb{N}$ with the cofinite topology to $\mathbb{R}$

Let $\mathbb{N}$ have the co-finite topology, and let $\mathbb{R}$ have the usual topology. Then what functions from $\mathbb{N}$ to $\mathbb{R}$ are continuous? I think the constant functions would ...
4
votes
1answer
78 views

Show $\lim_{m \to \infty ,n \to \infty } f(\frac{{\left\lfloor {mx} \right\rfloor }}{m},\frac{{\left\lfloor {ny} \right\rfloor }}{n}) = f(x,y)$

The question is Suppose $f(x,y)$ is defined on $[0,1]\times[0,1]$ and continuous on each dimension, i.e. $f(x,y_0)$ is continuous with respect to $x$ when fixing $y=y_0\in [0,1]$ and $f(x_0,y)$ is ...
5
votes
0answers
51 views

Continuity ( Functions of 2 variables ).

Given , $$ f(x,y) = \begin{cases} \dfrac{xy^{3}}{x^{2}+y^{6}} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \\ \end{cases} $$ We need to check whether the function is continuous at ...
1
vote
2answers
17 views

Extending continuous function from a dense set

If $X$ is a metric space and $Y$ a complete metric space. Let $A$ be a dense subset of $X$. If there is a uniformly continuous function $f$ from $A$ to $Y$, it can be uniquely extended to a uniformly ...
1
vote
2answers
32 views

Increasing and decreasing piecewise function on an interval

I'm working on a problem that involves finding the intervals where a function $f$ is increasing and decreasing. Given the function$$ f(x) = \cases{ x+7 & \text{if } x\lt -3\cr |x+1| & ...
2
votes
1answer
33 views

If $\ne: X \times X \to S$ is continuous, is X hausdorff?

The Sierpiński space is defined like so: $$S = (\{\top, \bot\}, \{\emptyset, \{\top\}, \{\top, \bot\}\})$$ (A nice way to visualize is to take [0, 1], and glue 0 on $\bot$ and (0,1] onto $\top$.) ...
1
vote
1answer
40 views

Using $\epsilon$-$\delta$ argument to show continuity

Show using the $\epsilon$-$\delta$ definition of continuity that $f(x)=\begin{cases} 11&\text{if}~0\leq x\leq 1\\x&\text{if}~ 1<x\leq 2\end{cases}$ is continuous on $[0,1)\cup (1,2]$ ...
-1
votes
0answers
26 views

Exercise of Application of Implicit Function Theorem

Let $f\colon U \subset \Bbb R^2 \to\Bbb R$ such that $$\forall (x,y) \in U \quad (x^2 + y^4)f(x,y) + (f(x,y))^3 = 1$$ Prove that $f$ is $C^\infty$.
1
vote
1answer
26 views

If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous, then $\|f(x)\|< \infty , x \in X$ and why? [closed]

If a function $f: X \to Y $ $X, Y $ are some normed spaces, is continuous. Then is $\|f(x)\|< \infty , x \in X$ and why? I am thinking yes on this one (because otherwise, it would not be ...
1
vote
3answers
54 views

Continuity of the inverse map

If we have a function $F(x): \mathbb{R^4} \rightarrow \mathbb{R^3}$. Defined as \begin{align} x_1\, x_4&=y_1 \\ x_2\, x_4&=y_2 \\ x_1^2+x_2^2-x_3^2&=y_3 \end{align} Can a continuous ...
0
votes
0answers
33 views

Counter example of non continuity

I present in the following a variation of the problem described in Continuity of a deterministic function generated from a probability function. There, it has been proved that $g(x)$ is not ...
-1
votes
0answers
16 views

Non-periodic continuous functions takes maximum and minimum at every compact subset of the domain?

Let $f:\mathbb R \rightarrow \mathbb R$ a continuous and periodic function, with period $p$. By Weierstrass's theorem, the restriction $f|_{[x_0,x_0+p]}$ is bounded and takes its maximum and minimum ...
0
votes
1answer
8 views

Why a function from a product of $\omega$-complete partial order is continuous?

In “The Formal Semantics of Programming Languages” by Glynn Winskel, there is Lemma 8.10 in “8. Introduction to domain theory/3. Constructions on cpo's/2. Finite products” which is promised to be ...
1
vote
1answer
25 views

Is this a Darboux function?

Let $f(x)=x$ if $0\leq x\leq 1$ and $f(x)=x-\frac{1}{2}$ if $1<x\leq 2$. This is a discontinuous function on $[0;2]$ but it satisfies the intermediate value theorem so it's a Darboux function. ...
3
votes
3answers
73 views

How to show a function is absolutely continuous?

Are all continuous functions also absolutely continuous functions or not? If it does, then does its inverse hold? Kindly give an example?
2
votes
2answers
31 views

Prove uniformly continuity at $\infty$ to continuous function

Say $f:[0,\infty) \to \mathbb{R}$ is a continuous function. Assume $\lim_{x \to \infty}[f(x)-ax]=b$ for some $a,b \in \mathbb{R}$ and prove $f$ is uniformly continuous in $[0, \infty)$ So if the ...
2
votes
5answers
222 views

Prove a function is not differentiable using continuity

Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$ Prove that the function is not differentiable at $x = \frac12.$ The answer in my book is as follows: $$\lim_{x \to \frac12-} ...
2
votes
1answer
59 views

Continuity of a deterministic function generated from a probability function.

I am working on the proof of a specific proposition on probability theory whose particular case for two variables is presented in the following. Let $X_1$ and $X_2$ be different random variables ...
0
votes
2answers
22 views

Continuous function's property proof using slightly different epsilon/delta definition.

I was asked to prove/disprove the following: If $f:\mathbb{R} \to \mathbb{R}$ continuous function, so for all $x \in \mathbb{R}$ and $\epsilon > 0$, exists $\delta > 0$ s.t. if $y \in ...
1
vote
1answer
44 views

The limit of a series of continous functions is continuous.

Given a continuous function $f_0: [0,1] \rightarrow \mathbb{R}$, define $$f_n(x) = \int^x_0 f_{n-1}(t) dt, x \in [0,1]$$ for $n=1,2,3,...$ . For each $x \in [0,1]$, show that ...
5
votes
2answers
54 views

Prove $ \ \frac{a}{x^3 + 2x^2 - 1} + \frac{b}{x^3 + x - 2} \ = \ 0 \ $ has a solution in $ \ (-1,1) $

If $a$ and $b$ are positive numbers, prove that the equation $$\frac{a}{x^3 + 2x^2 - 1} + \frac{b}{x^3 + x - 2} = 0$$ has at least one solution in the interval $ \ (-1,1) \ $ . The question is from ...
1
vote
1answer
43 views

What can be said about the continuous function $f:\mathbb R^{2} \rightarrow \mathbb R$ that has only finitely many $0$'s $?$

$f\colon \mathbb R^{2}\rightarrow \mathbb R$ is a continuous map that assumes $0$ for only finitely many points. Then which one is true A. either $f(x)\le 0$ for all $x$ or $f(x)\ge ...
0
votes
1answer
36 views

$f(x)=2-|x-3|, 1\le x\le 5$ and for other values, $f(x)$ is obtained using the relation $f(5x)=kf(x)$ for $x\in R$. then…

Question: The maximum value of f(x) in $[5^4,5^5]$ for $k=2$ is? Also, if $$\lim_{x\to \infty}\int_1^xf(x)dx$$ is a finite number, find the exhaustive set of $k$. Attempt : For first part, ...
1
vote
1answer
63 views

Exercise 43 chapter 2 in Real Analysis of Folland

I got stuck on this problem and couldn't find any clue to solve it. Can anyone give me some hint or give me some solution for it. I really appreciate! Suppose that $\mu(X) < \infty$ and ...
3
votes
5answers
240 views

Existence of solutions of the equation with a limit.

Let f be continuous function on [0,1] and $$\lim_{x→0} \frac{f(x + \frac13) + f(x + \frac23)}{x}=1$$ Prove that exist $x_{0}\in[0,1]$ which satisfies equation $f(x_{0})=0$ I suppouse that the ...
0
votes
3answers
48 views

The derivative function is not continuous

(Sorry about the bad title, couldn't think of a way to word it concisely.) Let $C[0, 1]$ be the metric space whose points are all continuous functions from $[0, 1] \rightarrow \mathbb{R}$ with the ...
3
votes
0answers
61 views

Why is $F$ continuous?

Why is the function: $F: P(\mathbb R) \to \mathbb R$, $F(X) = \int_X e^{-x} dx$ a continuous function? How to prove such a thing? Does it even make sense to talk about the continuity of such a ...
0
votes
3answers
25 views

Fixpoints and continuity

I don't understand why this is true: If $f:[0,1]\rightarrow[0,2]$ is a continuous function then exists $x \in [0,1]$ such that $f(x)=2x$ I don't understand why such a point exist. Why is there not ...
5
votes
2answers
43 views

On any continuous map $f:S^1 \to \mathbb R$

Let $f:S^1 \to \mathbb R$ be any continuous map , where $S^1$ is the unit circle in the plane . Let $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ ; then how to prove $A$ is uncountable , or ...
0
votes
1answer
17 views

Question on continuity and differentiability of min() and max() functions.

Question: $f(x)=x^2-2|x|$. Test the continuity of $g(x)$ in the interval $[-2,3]$ if $g(x)$ is defined as: attempt: $f(x)$ is defined as: But i am finding it difficult to understand $g(x)$. ...
2
votes
1answer
72 views

$|f(x)-f(y)| \geq \frac{|x-y|}{2}$

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function such that $|f(x)-f(y)| \geq \frac{|x-y|}{2}$ then prove $f$ is onto. I can prove it just using IVT, but looking for some short solution which ...
1
vote
1answer
18 views

Continuity of a parametrized surface integral of a sobolev function

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain and let $v\in H^1(\Omega)$. Furthermore, let $S=(0,T)$ denote a time interval and let $s\in ...
1
vote
2answers
90 views

Complex Analysis ( Limits at a point ).

We need to prove that $ \lim_{z \to z_{0}}(z^{2}+c)$ = $z_{0}^{2}+c$ , where c is a complex constant , using $\epsilon - \delta$ definition , where $z , z_{0}$ are complex variables. What I tried : ...
4
votes
1answer
103 views

Image of a continuous function

Let $f :\mathbb R \rightarrow\mathbb R$ be continuous function . Then which cannot be the image of $(0,1]$ ? A. $\{0\}$ B. $(0,1)$ C. $[0,1)$ D. $[0,1]$ Now A. is ...
0
votes
2answers
17 views

Question involving Taylor series and continuity

Question: $$f(x)=\lim_{n\rightarrow \infty}\frac{x^{2n}-1}{x^{2n}+1}$$ Where is this function continuous? Trial: I analyzed positive terms of x.For large values of n the function approaches to ...
1
vote
2answers
36 views

If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...
5
votes
1answer
184 views

Proving existence of at least one root

The function $f:\mathbb{R}\to\mathbb{R}$, is continuous and $a>0$. How can I prove that there is at least one root of this equation: $f(x)=f(\sqrt{|x^2-a|})$
-2
votes
0answers
39 views

How to prove $f:X\to Y$ is continuous

$X,Y$ are metric spaces. Then $f:X\to Y$ is continuous in $X$ if that $C\subset Y$ is closed implies that its inverse image is closed in $X$. I want a proof that's directly based on the ...
0
votes
2answers
55 views

Questions of an example of a measurable function fails to be continuous everywhere or even, almost everywhere

Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$. Definition of Lebesgue measurable function: Given a ...
7
votes
2answers
164 views

Proving the continuity of these maps

Backstory: I am having an exam soon, and these are the assignments that keep coming up, I cannot finish any of them to the end, but have ideas about solving them, and would like to hear your thoughts ...
14
votes
5answers
667 views

Why formulate continuity in terms of pre-images instead of image?

I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some ...
1
vote
2answers
287 views

Continuity Must Hold in an Entire Open Set?

Claim: If a function $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $\vec a \in \mathbb{R}^n$, it is continuous in some open ball around $\vec a$. Is this claim false? In other words, is it ...
2
votes
1answer
30 views

Uniform Continuity implies Continuity

Let $f$ be a function from a metric space $X$ to a metric space $Y$. Show that if $f$ is uniformly continuous on $X$ then $f$ is continuous on $X$. Show that the converse is not true. Uniform ...
0
votes
2answers
14 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
2
votes
1answer
50 views

Reverse Intermediate Value Theorem

What does it mean to say that a real valued function $ f : [a, b] \rightarrow \mathbb{R} $ is continuous at $ x_0 \in [a, b] $? Assume that $ f : [a, b] \rightarrow \mathbb{R} $ is continuous State, ...
0
votes
1answer
50 views

Discontinuities of an injective function from $\mathbb{R}$ to $\mathbb{R}$

It is well known that a monotonic function from $\mathbb{R}$ to $\mathbb{R}$ can have only countably many discontinuities. Question: Is it true that an injective function from $\mathbb{R}$ to ...