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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Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
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1answer
39 views

Compute the Supremum over all partitions of $[0,1]$ [closed]

Let $f \in C^1[0,1]$.For a partition $$(P):0=x_0 <x_1<x_2<...<x_n=1$$ define $$S(P)= \sum _{i=1}^{n} \vert f(x_i)-f(x_{i-1}) \vert $$ Compute the supremum of $S(P)$ taken over all ...
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1answer
43 views

Correctness of the proof that $ |f(c+h)-f(c)| ≤ w(|h|)$ implies $f$ is continuous at $c$

Question: Let $\omega: [0,\infty)\rightarrow [0,\infty)$ be continuous at $x = 0$ with $\omega(0) = 0$. Suppose for some point c $\in$ $\mathbb{R}$ the function $f$: $\mathbb{R}$ $\rightarrow ...
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2answers
44 views

Continuity of rational numbers

Given that a function $f: \Bbb R \to \Bbb R$ is continuous on $\Bbb R$ and that $f(r)=0$ $\forall r \in \Bbb Q$, I need to show that $f(x) = 0$, $\forall x \in \Bbb R$. Can anyone advise on how to ...
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1answer
30 views

Proof of Sequential Criterion

Here is my attempt at a proof for the Sequential Criterion. Definition: Sequential Criterion for Continuity A function $f: A \to \Bbb R $ is continuous at the point $c \in A$ if and only if ...
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0answers
13 views

Good reference for “solving equation $f(x)=0$ by homotopy and continuation methods”

I need a good reference for "solving equation $f(x)=0$ by homotopy and continuation methods". If $f:X\to Y$ is a continuous map between to linear space $X$ and $Y$, we want to find the roots of $f$. ...
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2answers
109 views

Diagnosing essential Classical Mathematical Analysis I knowledge needed for II

I need to take Classical Mathematical Analysis II (Chapters 7-10: Sequences & Series of Functions, Special Functions (Exp/Log/Fourier/Gamma), Functions of Several Variables, Integration of ...
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2answers
40 views

Which functions are $C^k$, but not $C^{k+1}$ on $ \mathbb{R} $

The only functions I can think of that fulfill this property are the polynomials of degree k. However this does not necessarily imply, that this are the only functions that are in such a space. So I ...
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1answer
47 views

Check the differentiablity of $\theta :\mathbb R^2 \setminus\{(0,0)\}\to \mathbb R$

For , $(x,y)\in \mathbb R^2$ with $(x,y)\not =(0,0)$ , let $\theta=\theta(x,y)$ be the unique real number such that $-\pi<\theta \le\pi$ and $(x,y)=(r\cos \theta , r\sin \theta)$ , where ...
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12 views

On existence of continuous mappings mapping open bounded path-connected sets onto open bounded path-connected sets

I was just thinking a bit and this question somehow arrived, I will ask it below after some preparation. Let us deal with this problem in $\mathbb R^n$. Now, suppose that we choose some two subsets ...
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3answers
46 views

If the sets $A=\{x\in E, f(x)<\lambda\}$ and $B=\{x\in E, f(x)>\lambda\}$ are open, then $f$ is continuous

I have that $f: (E,\theta)\rightarrow (\mathbb{R},|.|)$ an application, if we have that for all $\lambda\in \mathbb{R}$ the two sets $A=\{x\in E, f(x)<\lambda\}$ and $B=\{x\in E, f(x)>\lambda\}$ ...
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3answers
33 views

Continuity and uniform continuity

I'm having trouble understanding the notion of uniform continuity, the definition states as follows: Let $f: D\to\mathbb{R}$, $f$ is uniformly continous in $X\subset D$ if: $$\forall\varepsilon >0, ...
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0answers
36 views

Continuous function on connected set is constant

I have asked one question previously differentiable map on a connected open set which says : $f$ is differentiable on $E$ and $E$ is open, conencted and $f'(x)=0$ for every $x\in E$ then $f$ is ...
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1answer
16 views

If $V\cap f(X)\neq \phi$, $W\cap f(X)\neq \phi$, $f(X)\subseteq V\cup W$ and $f$ is continuous, is it true that $f^{-1}(V)\cup f^{-1}(W)=X$?

Let $$ f:X\rightarrow Y $$ be a continuous function. If $V\cap f(X)\neq \phi$ and $W\cap f(X)\neq \phi$ and $f(X)\subseteq V\cup W$, is it true that $f^{-1}(V)\cup f^{-1}(W)=X$? If it is true, ...
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2answers
35 views

If $f$ is defined on $X$ such that every compact set in $X$ is mapped to a compact set. Is $f$ a continuous map? [duplicate]

Let $$ f:X\rightarrow Y $$ If $f$ is defined on $X$ such that every compact set in $X$ is mapped to a compact set in $Y$. Is $f$ a continuous map?
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1answer
59 views

What is an example of a function that is continuous but not uniformly continuous? [duplicate]

I am trying to understand the difference between a continuous function and a uniformly continuous function. Is there example of a function that is continuous but not uniformly continuous and a ...
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0answers
17 views

Uniform convergeness and polynomials

If $f(x)$ is continuous function on $[a,b]\Rightarrow\exists <P_{n}(x)>$ sequence of polynomials such that $P_{n}(x)$ uniformly converges to $f(x)$ on $[a,b]$. Is this statement true or false? ...
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2answers
27 views

If the Continuous image of a space is compact, does that mean the space is compact? [closed]

Let $f$ be a continuous function and $f(X)$ is compact. Is $X$ necessarily compact? Is there an example to prove/disprove this? Thank you.
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3answers
34 views

If the pointwise limit is uniformly continuous, the functions in the sequence need not be so

If $f_{n}$ is a sequence of uniformly continuous functions and $f_n \to f$, then $f$ is a continuous function. Why is the converse of this statement not necessary true? Is there a simple example? ...
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5answers
82 views

Determine if a function is uniformly continuous by looking at its graph?

Today my professor introduced the idea of uniform continuity. However, I had difficulty visualizing what the graphical interpretation of a function being uniformly continuous is. Is it possible to ...
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48 views

Is this problem a particular case of some known theorem of Real Analysis?

Let $f \in C[0,2]$ is such that $f(0)=f(2)$ then there exist $x_1$ and $x_2 \in [0,2]$ such that $x_1 -x_2=1$ and $f(x_1)=f(x_2)$ The solution of this problem is easy one can just take the ...
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1answer
46 views

Evaluate $\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^nf'\left(\frac{k}{n}\right)$

Left $f:\mathbb{R}\to\mathbb{R}$ be continuously differentiable. We are to evaluate $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^nf'\left(\frac{k}{n}\right)$$. One thing I know is that $$\lim_{x\to ...
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1answer
25 views

Working with Ideas related to Continuity

I've been interested in figuring out, if $f^2(x) = f(x) \cdot f(x)$ is continuous, does that confirm that $f$ is continuous? We know that the domain between the two functions are the same, and so for ...
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1answer
57 views

Evolution of Definitions

I try to understand how the definitions of mathematics have evolved (or formulated)... I'll use the epsilon-delta continuity definition as an example to ask my question... It may seem trivial, but ...
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1answer
74 views

The limit of $(\sqrt{1+kx}-\sqrt{1- kx})/x$ as $x\to 0$ [closed]

For what value of k, $$f(x)=\begin{cases}\frac{\sqrt{1+kx}-\sqrt{1- kx}}{x} & \mbox{ if }-1 \le x <0 \\ \frac{2x+1}{x-1} & \mbox{ if } 0\le x<1\end{cases}$$ is continuous at $x= 0$.
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1answer
18 views

Function continuity problem

Let f be a continuous function in the interval $[a,\infty)$, and $\lim_{x \to \infty}f(x)=L$. Then f is bounded. I've been trying to prove by contradiction but I couldn't manage to prove that if it's ...
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0answers
18 views

Is breaks in the continuity of a graph scalar with regards to multiplying a constant?

I think I'm probably right but seeing how I've recently begun to go down the rabbit hole in mathematics that tends to produce all sorts of inconsistenties with any advanced conjectures that I make, I ...
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2answers
36 views

Proving Continuity On A Particular Function

I'm working on a problem where $S \subset \mathbb{R}$ arbitrary and I have a function $f(x) = \inf\{|x-s| : s \in S\}.$ I want to show that $f$ is uniformly continuous for all $x \in \mathbb{R}.$ I am ...
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0answers
31 views

Uniform continuity of $\cos x \cos \frac{\pi}{x}$ and $\sin x \cos \frac{\pi}{x}$ in interval $]0,1[$

I have to find the uniform continuity of $f(x)=\cos x \cos \frac{\pi}{x}$ and $g(x)=\sin x \cos \frac{\pi}{x}$ ,where $x \in ]0,1[$ My approach:- For $f(x)$- $$f(x)-f(y)=\sin ...
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1answer
40 views

Prove $S^1$ is not homeomorphic to $S^2$ using connectedness

I have to prove that the unit circle $S^1$ is not homeomorphic to the sphere $S^2$ using connectedness. Intuitively I know this is true, but I'm not sure how to prove this.. Can someone help me?
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3answers
82 views

Is $\sin^2x$ uniformly continuous on$x\in [0,\infty]$

I have the question that is $sin^2x$ uniformly continuous on $x \in [0,\infty]$ ? My approach: Let $\left|x-y\right|<\delta$ we have:- $$\left|sin^2x-sin^2y\right|=\left|(\sin x+\sin y)(sin x-sin ...
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23 views

Continuity properties of an example function $f:\mathbb{R}^n\to\mathbb{R}$

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ defined as follows: $$ f(x)=\begin{cases} ||x||^2 & \text{if $||x||\le 1$,}\\ 1/||x||^2 & \text{if $||x||> 1$,} \end{cases} $$ where ...
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0answers
16 views

Lipschitz condition and differentiability

I have a question related to Lipschitz condition and differentiability. Similar questions have been answered here and here but do not clarify all my doubts. CASE 1: Consider $f:\mathbb{A}\subseteq ...
4
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2answers
55 views

Multivariable Calculus problem: studying continuity of a function

I'm trying to solve the following problem: Let $\varphi : \mathbb{R}\to\mathbb{R}$ differentiable, and $\varphi'(x)$ continuous; $f:\mathbb{R}^2\to\mathbb{R}$ given by $\begin{equation*} ...
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0answers
28 views

Contractibility is a Weaker Notion than Deformation Retract to a Point [duplicate]

This is problem 6 in Chapter 0 of Hatcher's Algebraic Topology. Let $X$ be the subspace of $\mathbf R^2$ consisting of the horizontal line segment $[0, 1]\times \{0\}$ together with the vertical ...
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0answers
20 views

Is the second derivative of $\frac{1}{|x-y|}$ in $L^2(\mathbb{R}^3)$ or $L^1$?

I was unsuccessfully trying to show whether the function $\frac{\partial^2 }{\partial x_k \partial x_j}\frac{1}{|x-y|}$ for $x,y\in \mathbb{R}^3$ is in $L^2$ or in $L^1$? i.e if ...
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3answers
47 views

Continuity of a function from the reals to the reals that fixes rational numbers

Define $f:\mathbb{R} \to\mathbb{R}$ by $x\mapsto x$ if $x$ is rational and $x\mapsto 0$ if $x$ is irratoinal. Prove that $f$ is continuous at a point $a\in\mathbb{R}$ if and only if $a=0$. I'm ...
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142 views

if the inverse images of all closed balls are closed, is $f$ continuous?

Is the following statement true? (it is asked to be proved true) If $f: D \to\mathbb R^n$, and for every closed balls $B$ in $\mathbb R^n$, pre-image of $f$ of $B$ is closed in $D$, then $f$ is ...
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22 views

Does argsup function satisfy a property of the supremum

Let $X$ be a compact set of $d\times d$ matrices, and let $f\in C(\overline{\Omega})$, and $u\in C^2(\overline{\Omega})$. Define $A(x)=\operatorname{argsup}_{W\in ...
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continuity using epsilon delta definition

Can anyone tell me the answer using epsilon delta definition.
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2answers
41 views

$\varepsilon-\delta$ definition

I am trying to prove the continuity of $f \colon [0, \infty) \to \mathbb{R}$ $$f(x) = \frac{x^2}{x+1}.$$ I tried to use $|f(x) - f(x_0)|$ So: $$\left|\frac{x^2}{x+1} - \frac{x_0^2}{x_0+1}\right|$$ ...
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38 views

Let $H$ be a Hilbert space and $Φ≤H$ be equipped with a topology. Under which topology on $Φ^*$ is $H^*\ni f\mapsto\left.f\right|_Φ$ continuous?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $\Phi$ be a vector subspace of $H$ equipped ...
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2answers
47 views

Continuity at a point.

If the function $$F(x,y) = \frac{x^3y^3}{x^3+y^3}$$ continuous at $(x,y) = (0,0)$. It says not continuous at that point in my book but my answer is continuous. Here $F(0,0) =0$.
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12 views

Why can the definite integral of a function f from a to b be interpreted as the net sign area…?

between the graph of f and the interval [a,b], only if f is continuous? This is what my textbook mean by "net sign area": I don't think f has to continuous for this to be true. For example, f can ...
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1answer
22 views

Smoothness of division of infinitely differentiable functions

Suppose I have a $C^\infty$ function $f\colon\mathbb R\to\mathbb R$, $f(0)=0$, is it true that $g(x)=\frac{f(x)}{x}$ is also a $C^\infty$? If it is true, how do I prove it? Generalized to ...
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2answers
36 views

Prove a function is uniformly continuous

Prove the function $f(x)=\sqrt{x^2+1}$ $ (x\in\mathbb{R})$ is uniformly continuous. Now I understand the definition, I am just struggling on what to assign $x$ and $x_0$ Let $\epsilon>0$ ...
5
votes
1answer
58 views

If $f(0)=f(1)=1$ and $|f(a)-f(b)| < |a-b|$ then $|f(a)-f(b)| < \frac{1}{2}$

Problem: $f$ be a function on $[0,1]$ such that $f(0)=f(1)=1$ and $f(a)-f(b) < |a-b|$ for all $a$ not equal to $b$. Prove that $|f(a)-f(b)| < \frac{1}{2}$. My attempt: Things I observed are ...
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2answers
35 views

verification: did I proof continuity right?

Let $\mathbb{R}$ be endowed with the lower limit topology $\mathcal{T}_l$. That is, the smallest topology on $\mathbb{R}$ which contains all the intervals of type $[a,b)$ with $a,b\in \mathbb{R}$. Is ...
2
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0answers
55 views

compact set in a space of functions continuous in $\mathbb{R}$

As it is known the set $\mathcal{B}=\{ f:\mathbb{R}\rightarrow \mathbb{R} : f \mbox{ is continuous}\}$ it is not a metric space with the metric $d(f,g)=sup_{x\in \mathbb{R}}\| f(x)-g(x)\|$ it can ...
4
votes
1answer
58 views

Finding $f\in C( \mathbb R)$ such that for some integer $n>1$, $f^n(x)=x,\,\forall x \in \mathbb R$

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that for some integer $n>1$, $f^n(x)=x,\,\forall x \in \mathbb R$; then is it true that either $f(x)=x,\,\forall x \in \mathbb R$ or ...