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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58 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
20 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 ...
1
vote
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?
4
votes
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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0answers
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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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
votes
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 ...
0
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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 ...
10
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2answers
145 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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0answers
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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25 views

continuity using epsilon delta definition

Can anyone tell me the answer using epsilon delta definition.
2
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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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0answers
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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votes
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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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
23 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
37 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 ...
0
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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 ...
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3answers
42 views

Show that the collection of open balls in two metric spaces are identical

I am having trouble trying to prove the following statement. I can see why it would be true intuitively, however, I am having trouble formalising the proof as the notation is quite confusing. Show ...
1
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1answer
32 views

Closed, continuous, surjective map with inverse images compact

Old qual question: Let $p:X\to Y$ be a closed, continuous, surjective map such that $p^{-1}(y)$ is compact for every $y\in Y$. Let $(U_\alpha)_{\alpha\in A}$ be an open cover of $X$. Show that any ...
3
votes
1answer
18 views

which $p,q$ makes the following function continuous

Let $p,q > 0$ and $\max \{ p,q \} > 1$. Let $ f(x) = \sum _ { n =1 } ^ \infty \frac { x } { n^p + x ^2 n ^q } $. The problem asks which $p,q $ makes $f$ continuous on all of $\mathbb R$. I know ...
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0answers
28 views

What does notation $C^{\beta}[0,1]$ mean?

What does the notation $C^{\beta}[0,1]$ for $\beta \in (0,1]$ mean? I know $C[0,1]$ is the space of all continuous functions on the interval $[0,1]$, but what about $C^{\beta}[0,1]$? Usually ...
4
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1answer
37 views

Show that for each $0<r<1$ there is such $x,y \in [0,1]$ s.t. $|x-y|=r$ and $f(x)=f(y)$.

let $f:[0,1] \rightarrow \mathbb R$ be continuous and non-negative.its given that $f(0)=f(1)=0$. show that for each $0<r<1$ there is such $x,y \in [0,1]$ s.t. $|x-y|=r$ and $f(x)=f(y)$. ...
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2answers
80 views

Counterexample to: “If a function is continuous in a point $x_0$ then it is defined in a neighborhood of that point.”

I'm looking for a counterexample to the statement: If a function is continuous in a point $x_0$ then it is defined in a neighborhood of that point. If I take $f: \mathbb{R} \setminus ...
3
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0answers
42 views

Example of a function differentiable in a point, but not continuous in a neighborhood of the point?

Is there a function that is differentiable in a point $x_0$ (and so continuous of course in $x_0$) but not continuous in a neighborhood of $x_0$ (as said, besides the point $x_0$ itself)? Can anyone ...
3
votes
1answer
63 views

On preimage of open sets of functions on real line having at most countably many discontinuity points

Let $f:\mathbb R \to \mathbb R$ be a function whose set of discontinuity points is at most countable ; is it true that for every open set $G \subseteq \mathbb R$ , there is an open set $U$ and a ...
3
votes
1answer
25 views

Determine if a function is the null function from conditions on integral

I want to find out if the following statement is true and if it is to prove it: for $f : [a, b] \rightarrow \mathbb R$ continuous and integrable, if $\int_a^b f(x) dx = 0$ and $\int_a^b x f(x) dx = ...
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1answer
36 views

'Continuity' necessary in proof dynamical systems?

The following is a (rough) translation of a statement and proof given during a course in dynamical systems. Let $D \subseteq \mathbb{R}^2$ be and open set, $f: D \to \mathbb{R}: (t, x) \mapsto ...
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Local extrema proof by contradiction?

I have a continuous function $f$ over an interval $\left [ a,b \right ]$ such that $f(a)=f(b)$. Also $f$ admits no local extrema over this interval. I can say that this function reaches a maximum ...
2
votes
1answer
40 views

Classifing a singularity.

I have the function: $$f(z)=z \cos\left(\frac{1}{z}\right)$$ which has a singularity at $z=0$ and $f(z)\to 0$ as $z\to0$. The theory says that a limit implies that the singularity is removable. But ...
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0answers
29 views

Continuity after taking partial inverse

Suppose $f(x,y)$ is jointly continuous and strictly increasing in both $x$ and $y$. For fixed $y$, let $f_y^{-1}$ be the inverse of $f(\cdot,y)$. How can I show that, for a fixed $x$, the function $y ...
0
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1answer
17 views

Polar coordinates approach to find continuity at a point

I have to investigate continuity at (1,2) of the following function:- $$f(x,y)= x^2+2y, (x,y)\ne(1,2) ;$$ $$f(x,y)=0 ,(x,y)=(1,2)$$ My approach:- I have considered a circle of radius r around ...
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1answer
20 views

What method can I use to determine continuity of squares on a 2d grid?

I have N squares aligned to a 2d grid. I'd like to know if the set is continuous -- that is to say, that each of the N squares is adjacent to at least one other square in the set. This is quite ...
12
votes
1answer
197 views

Countable-infinity-to-one function

Are there continuous functions $f:I\to I$ such that $f^{-1}(\{x\})$ is countably infinite for every $x$? Here, $I=[0,1]$. The question "Infinity-to-one function" answers is similar but without the ...
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0answers
48 views

Visit probability as a function of continuous time

I am working on a project aiming to model visit probabilities in spacetime prisms. On a given location, I know the visit probability at any time (within the prism boundaries), i.e. the visit ...
10
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1answer
411 views

Infinity-to-one function

Are there continuous functions $f:I\to S^2$ such that $f^{-1}(\{x\})$ is infinite for every $x\in S^2$? Here, $I=[0,1]$ and $S^2$ is the unit sphere. I have no idea how to do this. Note: This is ...
2
votes
1answer
27 views

Verification of proof of continuity between metric spaces and deduction from proof

Let $M = [0,1]^{[0,1]}$ and $d(f,g) = \sup{\{\lvert f(x) - g(x)\rvert \mid x \in [0,1]\}}$. For $a,b \in [0,1]$ let $\phi_{a,b}(f) = f(b) - f(a)$ ($\phi$ maps from $M \to \Bbb{R}$). Assume that ...
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vote
0answers
36 views

Prove that $\phi : C[a,b]\rightarrow \mathbb{R}$ given by $\phi(f)=\int_a^bfdx$ is uniformly continuous

Prove that $\phi : C[a,b]\rightarrow \mathbb{R}$ given by $\phi(f)=\int_a^bfdx$ is uniformly continuous. First, since $f$ is continuous $\phi$ is well defined (integrals exist). Since $f$ is ...
0
votes
1answer
54 views

$X,Y$ be connected ; $f:X\to Y$ be a continuous function which is right-cancellative w.r.t. continuous maps on connected spaces ; is $f$ surjective? [closed]

Let $X,Y$ be connected topological spaces and $f:X\to Y$ be a continuous function such that for any connected space $Z$ and any continuous functions $g_1,g_2:Y \to Z$ , $g_1 \circ f=g_2 \circ f ...
2
votes
2answers
83 views

Where is the given function discontinuous?

Where is the given function discontinuous? $$f(x)= \begin{cases} x^2&\quad\text{if }x < -1\\ \sqrt{x + 4}&\quad\text{if }−1 ≤ x ≤ 0\\ \sin(2x)/x&\quad\text{if }x > 0\\ ...
7
votes
2answers
173 views

Continuous map and irrational numbers

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous map such as each irrational number is mapped to a rational number (i.e. ...