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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5
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What is the domain of $f(x)=\frac{1}{x}-\frac{1}{x}$?

Question: What is the domain of $f(x)=\frac{1}{x}-\frac{1}{x}$? Does the function have a removable discontinuity at $x=0$? My attempt: My first intuition told me that it was $\mathbb R$, since we ...
2
votes
2answers
44 views

Function on half plane, continuity

let $\mu$ be a finite positive borel measure on $\mathbb{R}$ and let $\mathbb{H}$ denote the upper half plane $\{(x,y) \in \mathbb{R}^2: y > 0\}$. consider the functions ...
3
votes
3answers
39 views

How to show $\sqrt{|x|}$ is not Lipschitz continuous?

$f(x) = \sqrt{|x|}$ is a famous example of a function which is not Lipschitz continuous but is uniformly continuous. This link shows detailed explanation of it. Here provides the figure of this ...
7
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2answers
109 views

Proving that the product of two numbers (in $\mathbb{R}$ or $\mathbb{C}$) is a continuous function.

This is what is given in the textbook, I will highlight what is confusing me: Product in field $\mathbb R$ or $\mathbb C$,on $X \times X$ defined as: $$(x,y)\mapsto xy$$ (Let indicate that map with ...
1
vote
1answer
39 views

“continuously differentiable $\subseteq$ Lipshitz continuous” with $f(x) = x^2$

In the Wiki, it says: continuously differentiable (i.e. class $C^1$) $\subseteq$ Lipshitz continuous. Consider the simplest example ($x,y\in \mathbb{R}$): $$f(x) = x^2$$ It is not Lipshitz ...
2
votes
2answers
29 views

Function of several variables which is continuous at single point

Examples of functions on $\mathbb{R}$ which are continuous at a single point are well known. But what about $f:\mathbb{R}^2\to \mathbb{R}$ which is continuous at a single point? I tried to proceed as ...
0
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3answers
46 views

Proof that |x-a| is continuous at x=a (epsilon delta), and nondifferentiable at x=a.

I need help justifying that $|x-a|$ is continuous and non-differentiable at $x=a$. I would also like to prove that it achieves a minimum at $x=a$, but I do not know if that is already clear enough.
2
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0answers
23 views

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where…

How does one prove that $ C_{\mathbb P} (\mathbb{I_r})$ is closed where $$\mathbb{I_r}=[x_0-r,x_0+r]$$ and $$\mathbb{P}=\{(x,y): |y-y_0|\leq a, |x-x_0|\leq b\}\subset \mathbb G $$ where $\mathbb G-$ ...
3
votes
2answers
67 views

Why do we care if a function is uniformly continuous? [duplicate]

There are a lot of question regarding whether a function is or is not uniformly continuous or just continuous and there are a lot of $\epsilon_s$ and $\delta_s$ trying to show whether a function is ...
1
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1answer
41 views

Decide if the following functions are not continuous on $(-\infty, \infty)$

Suppose $g(x)$ is continuous on $(-\infty, \infty)$. Determine if the following functions are or are not cont. on $(-\infty, \infty)$ and explain. a) $k(x) = \frac{x^2}{4 - (g(x))^2}$ b) $j(x) = ...
2
votes
1answer
38 views

Non injective continuous maps

Motivated by comments on this question we ask the following question: Let $f:M\to M$ be a continuous map where $M$ is a compact manifold and $f$ is not injective. Are there necessarily ...
1
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1answer
44 views

A continuous function from $\mathbb R\to \mathbb R^2$

Is there a continuous function $f:\mathbb R\to \mathbb R^2$ such that $f(\cos n)=(n,\frac{1}{n})$ for all $n\in \mathbb N$? I think this is not possible as if $f$ is continuous then the function ...
0
votes
0answers
35 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
1
vote
2answers
58 views

Why is $f(x) = x^2$ uniformly continuous on [0,1] but not $\mathbb{R}$

According to How exactly can't $\delta$ depend on $x$ in the definition of uniform continuity? There is a lot of agreement that $x^2$ is not uniformly continuous. But is $x^2$ uniformly ...
6
votes
3answers
85 views

Prove that $f'(0)=L$.

Let $f$ be continuous at $0$. Suppose lim$\displaystyle _{x\rightarrow 0} \frac{f(2x)-f(x)}{x} =L$. Prove that $f'(0)=L$. My Work: $\displaystyle ...
1
vote
0answers
25 views

A question on continuity of a piecewise function with 4 constants

I have this function, and I need to find the values of $a, b, c$ and $d$ so that $f(x)$ will be differentiable everywhere. $$f(x)=\begin{cases} ax+b, & x<-2 \\ x^2+c, & -2\le x\le2\\ ...
1
vote
1answer
42 views

Show that the sequence does not converge

My Try: $|f'(a)|>1$. Assume that the sequence converges to a limit $b$. Then $f(b)=b$. Since $a$ is the only fixed point it implies that $b=a$. Hence, given any $\frac{1}{m}$ where $m\in ...
0
votes
4answers
73 views

Proving that a continuous $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A) \text{ connected}$

Proving that $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A)-\text{connected}$ The answer is given like this just one step I do not understand ...
3
votes
2answers
76 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
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votes
0answers
36 views

Smoothing a function [closed]

Can you smooth a non-smooth function by: Differentiating it until you get a non-continuous function Changing that derivative to make it continuous by replacing the portions where there are jumps by ...
0
votes
1answer
14 views

Continuous function space and Reproducing kernel Hilbert [closed]

Let $E=C[-1,1]$, space of all real-valued continuous functions on [-1,1], $E$ is a reproducing kernel hilbert space? by inner product $\int_{-1}^{1} f(x)g(x) w(x) dx$ where $w(x)>0$ is weighted ...
1
vote
1answer
28 views

Question about continuity of piecewise function of two variables

Let $$ f(x,y)= \left\{ \begin{array}{ll} \left(x\sin\left(\frac{y}{x}\right),\frac{\cos (y) -1}{y}\right) & x \neq 0 \wedge y \neq 0 \\ (0,0) & x = 0 \vee y = 0 \\ \end{array} ...
0
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2answers
31 views

Piecewise $\mathscr C^1$ and piecewise continuous

I'm a little bit confused in piecewise continuity of a function. Say, if we have an odd function like $f(x) = x$ defined on the open interval $(0, \pi)$. We then extend it to a period $2\pi$ function ...
5
votes
2answers
141 views

continuity of a function

I have a task as preparation for my Calculus Exam. $f(x)= \begin{cases} 2^{\frac{1}{x-2}} ,& x\neq 2 \\ 0 ,&x=2 \end{cases}$ Now we have the following solution by one of our tutors: $l_1 = ...
1
vote
1answer
23 views

Showing that a continuous function is greater than zero

I've been working on a problem that wants me to show that given a function $f$ that is continuous at the point $c$ that, $$f(c)>0 \to \exists \delta\;\ \text{such that}\;\ f(x)>0\; \forall x ...
-1
votes
1answer
34 views

Proving that is $A:X \implies Y$ is a linear operator from metric space X to Y is continuous iff it is bounded bounded

The $\implies$ part interests me. The proof given goes like this: Let $A$ be continuous in 0 (because the 0 vector is in every vector space) $B_y(0,r)=\{y \in Y | \| y\|<r \} \implies \exists ...
0
votes
0answers
14 views

is this multivariable function twice continuously differentiable with respect to the parameter?

I have the following function $V: {R}^{*}_{+} \times {R}^{*}_{+} \rightarrow R$ , a and b are strictly positive real coefficients: $$V ( x_i(l) , x_j (l) ; l ) = a x_i (l) - b x_i (l)^2 + l^2 x_i ...
2
votes
1answer
39 views

Show continuity using epsilon delta definition for piecewise function [closed]

Using epsilon delta definition, show that $g$ is continuous on the whole of $\mathbb R$ $$g(x)=\cases{x^2 & \text{ if } x<1\\ \sqrt{x} & \text{ if } x≥1.}$$
2
votes
1answer
56 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
1
vote
0answers
35 views

Continuity by composition with a homeomorphism

I only want to know what do you guys think about the following proof. That's an exercise I've tried to do and I don't have an available answer, so... If you find some error or imprecision, I'd be ...
0
votes
0answers
14 views

continuous random variable - pth percentile

Let X be a loss random variable with cdf $$ F(x) = \left\{ \begin{array}{ll} 1-(θ/θ+x)^α & \textrm{for $x≥0$}\\ 0 & \textrm{for $x<0$}\\ \end{array} \right. $$ The 10th percentile is θ−k. ...
1
vote
1answer
45 views

Proof of the continuity of a function at irrational points

The problem is to prove that, If $f:\mathbb{R}\to\mathbb{R}$ defined by $$ \begin{align} f(x) = \begin{cases} 0 & \text{if $x\in \mathbb{R}\setminus\mathbb{Q}$}\\ \dfrac{1}{n} & \text{if ...
0
votes
1answer
20 views

(Just for clarification) - Is a convex, piecewise continuous function f on an closed interval continuous?

Lets say f is defined on an Interval $I = [a,b] $. Since f is convex, one immediately knows that f is continuous on $I^°$ , however left are the points $a$ and $b$ The piecewise continuity of f ...
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votes
0answers
41 views

Continuity of a real function on interval [closed]

I am having a question about the continuity of a real function which is given below: Is the following question true, if yes then what is the solution for it. Let f be a function which takes a real ...
2
votes
1answer
22 views

Prove continuity for a given norm

I struggle with this exercise from an analysis 2 book I use for self study: Let V := $C^1([0,1]; \mathbb{C})$ the vector space of continously differentiable functions from $[0,1]$ to $\mathbb{C}$ ...
6
votes
1answer
82 views

Is there continuous $f: [0, 1] \rightarrow [0, \infty)$ such that for all $x$ there is $y$ with $f(y) < f(x)$?

I think there isn't. Here's a sketch of a proof. I'm just not sure whether it really works because I'm not confident with the transfinite versions of the standard theorems about limits and convergent ...
-1
votes
2answers
51 views

Why the length of the zigzag curve approximating the circle does not approach the length of the circle?

I recently bumped into this question which asks why $\pi=4$ is wrong. And some answers(see the answer of user TCL, for example) stated that this has to do with functions and their derivatives. ...
4
votes
4answers
361 views

Why do Topologies get “finer”?

Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that ...
3
votes
1answer
31 views

Prove continuity of averaging function for integrable $f$

I want to prove the following statement which is part of a lemma in my textbook: Suppose $f$ is integrable on $\mathbb{R}^n$ and $x$ be a lebesgue point of $f$. Let $$M(r)=\frac{1}{r^d}\int_{|y|\le ...
1
vote
1answer
25 views

Evaluation function is Lipschitz wrt uniform conv metric

In the book on Brownian motion by Schilling and Praetzsch there is following statement: Let $\mathcal{C}_{(0)}:=\{f\in\mathcal{C}[0,\infty):\ f(0)=0\}$ be the space of all continuous functions ...
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votes
1answer
64 views

Continuity of a function on $\Bbb R^2$ [closed]

Function $f(x,y)$ is defined in a neighborhood of $(0,0)$. Then if for any t function $g(x) = f(x,tx)$ is continuous at $0$, then $f$ is continuous at $(0,0)$. if $f$ is continuous at ...
6
votes
1answer
90 views

Continuous map $\mathbb{R}^n\rightarrow\mathbb{R}^n$

When we say some map $\phi=(\phi_1,\ldots,\phi_n)$ is a continuous map $\mathbb{R}^n\rightarrow\mathbb{R}^n$ we really mean that each component $\phi_i$ is continuous as a function ...
1
vote
1answer
77 views

Need to prove continuous periodic function of $\varphi (x) \equiv \psi(x)$

Question: Let two $\varphi(x) $ and $\psi(x)$ periodic and continous functions such that $$ \lim_{ x\to\infty}(\varphi(x)-\psi(x))=0, \quad x\in \mathbb{R}. $$ Prove that $$ ...
3
votes
2answers
70 views

Integrating over a somewhat continuous function

I have a function $q(t)$ that starts at $q(0)=q_0$ and needs to get to $q(1)=q_1>q_0$. I have a free parameter $z$ that I can wiggle around to control $q'(t)$. Namely, I have a function ...
2
votes
1answer
57 views

Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?

Is it true that for every compact subset $A$ of $\mathbb R^2$ , there exist a compact set $B$ in $\mathbb R$ such that there is a continuous surjection from $B$ to $A$ ?
3
votes
3answers
105 views

Which statement “must be false”?

Given a function $f$ continuous on $[-4, 1]$ with its maximum at $(-3, 5)$ and its minimum at $(1/2, -6)$, is it not correct to say that both statements (B) and (D) must be false? (A) The graph of ...
4
votes
1answer
63 views

Show that $f(x) = \cos(2x)$ is uniformly continuous on $[0,\infty)$

Let $f: [0,\infty) \to \mathbb{R}$ and let $f(x) = \cos(2x)$. Show that $f(x)$ is uniformly continuous on $[0,\infty)$ Mt attempt: We have, $\forall \epsilon >0, \exists \delta > 0, s.t.\mid ...
2
votes
1answer
54 views

Continuity at $x=0$ of this function

Not a hard exercise:$$f(x)=\frac{1}{x^3}\cdot \int_{-x}^x \sin(4t^2) \, \text{d}t \quad \text{where} \space x\ne 0\:$$ $$f(x)=5\:;\:x=0\:$$ Checking it's continuity at $x=0$ by using L'Hospital's ...
0
votes
1answer
54 views

2 exercises: finding the limit and showing continuity and differentiability

part 1: $$\lim _{x\to _{x\to \frac{\pi }{2}^{-\:\:}} }\left(tg\left(x\right)\right)^{\sin\left(2x\right)}$$ so if $$\lim _{x\to _{x\to \frac{\pi }{2}^{-\:\:}} ...
3
votes
2answers
56 views

Prove that $f_A (x) = d({\{x}\}, A)$, is continuous.

Prove that: Let $(X, d)$ be a metric space, and let $A$ be a subset of $X$. The function $f_A\colon X\rightarrow \mathbb{R}$, defined by $f_A (x) = d({\{x}\}, A)$, is continuous. Honestly, I ...