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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Limit calculation and discontinuity

Having a function, which has a polynomial in the denominator like: $$ \lim_{x \to 2}\,\dfrac{x+3}{x-2} $$ We see there is a discontinuity at x=2, because it sets the denominator to 0. But ...
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2answers
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Spectral Measures: Support vs. Norm

Given a complex Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: $$T:=\int_\mathbb{C}zdE(z)$$ ...
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2answers
39 views

Continuity of a mapping $C\to C^2$, $C$ being the Cantor set

I will denote the Cantor set as $C$. We have proved earlier that every $x\in C$ can be uniquely written in a ternary representation $x=0.a_1a_2a_3...$ where all the $a_i \in \{0,2\}$. Now we consider ...
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1answer
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+50

Lipschitz continuity of parametric optimizer

Consider the parametric optimal solution $x^{*}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined as $$ x^*( y ) := \arg\min_{x \in X } \ \ x^\top x + x^\top A y \\ \quad \qquad \text{subject to: } \ ...
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1answer
218 views

A function having limit at every point but continuous nowhere

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?
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Newton method and the Banach fixed-point theorem

I try to combine the Newton method and the Banach fixed-point theorem but I have still some questions: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. ...
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1answer
23 views

Calculus: Continuous/Differentiable

I have no idea on how to do this problem. $$f(x)= \begin{cases} 2x^2-3x+1& \text{x<1}\\ (x-1)^{\frac{3}{2}}& \text{x $\geqslant$ 1} \end{cases}$$ a. Show that $f$ is continuous at $1$. ...
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3answers
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Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$

Suppose that $f: \mathbb R \to\mathbb R$ satisfies $f(x + y) = f(x) + f(y)$ for each real $x,y$. Prove $f$ is continuous at $0$ if and only if $f$ is continuous on $\mathbb R$. Proof: ...
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2answers
68 views

Two continuous functions with connected images

Suppose we have two continuous functions $f(x)$ and $g(x)$. Define $f$ on $[0,1]$ and $g$ on $[1,2]$, such that $f(1)=g(1)$. If we know that $\text {Im} (f(x))$ and $\text{Im} (g(x))$ are connected, ...
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1answer
35 views

Continuity vs differentiability [on hold]

If a derivative is increasing on a given interval, is it then also continuous on that interval? I.e. $f'(x)$ is increasing on $[a,b]$. Is $f'(x)$ continuous on $[a,b]$?
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Is there a valid multiplication for any choice of identity in $C(\mathbb{R})$?

Let $C(\mathbb{R})$ be the ring of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Its identity with the usual multiplication is $1(x) = 1$. I have two related questions. Firstly, when we ...
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1answer
20 views

Necessity in Arzela-Ascoli theorem

I am trying to prove necessity of boundedness and equicontinuity in Arzela-Ascoli and I don't know how to go about it. More precisely,I have: Let $K$ be a compact metric space, and $A\subset C^0(K)$ ...
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1answer
16 views

If f is a real function, continuous at a and f(a) < M, then there is an open interval I contianing a such that f(x) < M for all x in I.

Can someone please help? If f is a real function which is continuous at a ∈ R and if f(a) < M for some M ∈ R, prove that there is an open interval I containing a such that f(x) < M for all x ∈ ...
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1answer
15 views

Define $f(y)=d(x_0,y)$, prove that $f$ is continuous.

Consider a metric space $(X,d)$ and some $x_o \in X$. Define function $f_{x_0}(y)=d(x_0,y), $ which is in $\text{R}$. Show that the function is continuous. Here's what I've tried: According to ...
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3answers
33 views

Is set of all contiuous functions subspace?

This is one of the problems from the book: Hoffman and Kunze, chapter: Vector Spaces Let V be the (real) vector space of all functions f from R into R. Is the set of all f which are continuous, ...
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1answer
21 views

prove that $f(x,y) = x^2+y^2$ is continuous on rectangle R.

where $R = \{(x,y): |x|, |y| \leq \frac{1}{\sqrt 2} \}$ I am trying to use picard's theorem so I have to prove that f is continuous on R and that it's lipschitz continuous. How would I do this? I ...
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3answers
31 views

example of two continuous real-valued functions whose product is 0

Is there an example of two continuous real-valued functions, say on some interval, whose product is 0?
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2answers
32 views

Unbounded function on compact interval?

So what are some unbounded function on compact interval, if there is any? Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?
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1answer
75 views

Prove that $f(x)$ is a constant function.

Here is the question: Let f be a real valued continuous function on $[0, ∞)$. Suppose $f (x) = f (x^2)$ for all x ≥ 0, prove that f (x) is a constant function. My attempt: Since f(x) is continuous, ...
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2answers
27 views

The plane minus the graph of a continuous function consists of two path-connected components?

Let $f:\Bbb R\rightarrow \Bbb R$ be continuous. Show that $\Bbb R^2-\mathrm{graph}(f)$ consists of two path-connected components. I can show that the area 'above' the graph of $f$ and the area ...
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1answer
25 views

If $f$ is continuous on a bounded closed interval, then the supremum of $|f|$ is finite

If $f \colon [a,b] \to \mathbb{R}$ is continuous, then $\sup_{x ∈ [a,b]}\left | f(x)\right |$ is finite. Attempt: Suppose $f\colon [a,b] \to \mathbb{R}$ is continuous, then by the Extreme value ...
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1answer
26 views

What are the continuous functions that satisfy the following?

$f(x) = \begin{cases} 0, & x < 0 \\ 1 - f\left(\dfrac{1}{x}\right), & x > 0\text{.} \end{cases}$ I want this to generate a random variable that will be used as a proportion in a way ...
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1answer
16 views

Distance to a set

I have a question concerning to the following problem. Let $(X,d)$ be a metric set. For every subset $T \subset X$ we define a mapping \begin{equation} d_T : X \rightarrow R , d_T(x) := inf\{d(x,y) | ...
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1answer
17 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
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1answer
276 views

Is the infimum of a continuous function reached?

I am stuck on this problem for a while now, any help would be appreciated. I am working on the proof of a Network Calculus theorem, and I would like to show that the infimum is reached in the ...
3
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1answer
47 views

Show that $f$ is continuous if it follows the intermediate value property

If $f: [a,b] \to \mathbb{R}$ is $1-1$ and has the intermediate-value property -- that is, if $y$ is between $f(u)$ and $f(v)$, there is at least one $x$ between $u$ and $v$ such that $f(x)=y$ -- show ...
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5answers
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Looking for an example of a bijective continuous function $f:\mathbb{Q} \to \mathbb{Q}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$?

Clearly such a function does not exist from $\mathbb{R}$ to itself, but apparently it does in $\mathbb{Q}$ and I don't see how it could... Can you give me an example and explain to me how you thought ...
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1answer
39 views

A function that maps irrationals to rationals and rationals to irrationals cannot be continuous [duplicate]

I have a problem from my friend: If $f: \mathbb R \to \mathbb R$ maps every irrational number to a rational number, then $f$ is not continuous. (Also, $f$ has to map rational numbers to irrational ...
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1answer
14 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
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3answers
192 views

Uniform continuity of continuous function on a subset

Assume that $f: \mathbb R \rightarrow \mathbb R$ is continuous on the compact set $A$. Does for any $\varepsilon >0$ exist a $\delta >0$, such that $$ \lvert\, f(x)-f(y)\rvert<\varepsilon ...
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2answers
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Bounded - Continuous Relation

How to solve the following question? $$$$ Suppose $f:A\subset\Bbb{R}^2\to\Bbb{R}$ continuous in the rectangle $A=\{(x,y)\in\Bbb{R}^2|\alpha\leq x\leq\beta;\alpha'\leq y\leq\beta'\}.$ Proof that $f$ ...
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3answers
131 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
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3answers
105 views

Fundamental limit in two variables

Can I write that $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{u\to0}\frac{\sin(u)}{u}$$ and, hence, that $\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=1$? If so, why can I do it?
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Continuous functions satisfying a condition to be convex

Let $f$ be continuous on $\mathbb R$, and satisify $$f(x)\leq \frac{1}{2h}\int_{-h}^h f(x+t)d t, \forall\ h>0.$$ Show that $f$ is convex. The original question is "if and only if". However, I ...
4
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1answer
64 views

Show that the function is discontinuous in $\mathbb{R}$

"Show that the function $f: \mathbb{R} \rightarrow \mathbb{R}$, $$f(x)=\lim_{m \rightarrow + \infty}{ \lim_{n \rightarrow +\infty}{(\cos{(m! \pi x)})^n}}$$ is discontinuous at each $x \in ...
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1answer
37 views

Continuous function with continuous one-sided derivative

Simple example of the absolute value function $x \mapsto |x|$ on $\mathbb{R}$ shows that it is possible for a continuous function to posses both the right-hand and the left-hand side derivatives and ...
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1answer
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Continuity with restrictions

Suppose that $f \colon A \to \mathbb{R}$ is a function and that $B \subseteq A$. We define the restriction of $f$ to $B$ to be the function $f|_B B \to \mathbb{R}$ defined by $f_B(x) = f(x)$ for all ...
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3answers
29 views

the map $f:[0,1]\to [a,b]$ $f(x,y)=(1-x)a+xb$ is a homeomorphism

A question I just came across : A bijection $f:X\to Y$ is a homeomorphism if $f$ and $f^{-1}$ are continuous . Show that the map $f:[0,1]\to [a,b]$ $$f(x)=(1-x)a+xb$$ is a homomorphism... I ...
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1answer
47 views

Circle Equation Surjectivity

Consider the circular function $g:\mathbb{R}^{2} \to \mathbb{R}^{+}$, $g(x,y)=x^{2}+y^{2}$. Show that it is surjective and continuous. Note This post has been amended in accordance with the ...
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A continuous function on a finite union of closed intervals $X$ s.t. $f(x)\neq x$ for all $x \in X$

Let $X$ be a finite union of closed intervals of $\mathbb{R}$, and let $f: X \rightarrow X$ be a continuous function on $X$ such that $f(x)\neq x$ for all $x \in X$. What would an example of such a ...
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1answer
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Composition of two continuous functions is continuous

Let $f, g$ be functions $f$ is continuous at a, $\operatorname{f}(a) = b \in \operatorname{Dom}(f)$, $g$ is continuous at b. Then $g\circ f$ is continuous at $a$. Proof: Let $\varepsilon>0$, ...
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1answer
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Doubt on understanding continuity .

Just preparing for my multivariable-calculus exam and wanted to clear these things: I've come across many questions of sort below ,especially 2-dimensional regions, and wanted to understand the ...
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2answers
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Prove that a function is continuous in R^2

Prove that $f$ is continuous at $(0,y_0)$ where $f$ is defined on $\mathbb{R}^2$ by $$ \begin{cases} (1+xy)^{1/x} & x\neq 0 \\ e^y & x=0 \\ \end{cases} $$ I'm not really ...
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2answers
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Continuity of $f$ on $\mathbb R^2$

The question says: Let $$f(x,y) = \begin{cases} \dfrac{\text{sin}^2(x-y)}{|x|+|y|} & \text{if $|x|+|y|>0$} \\ 0 & \text{if $|x|+|y|=0$} \end{cases}$$ Is $f$ continuous on $\mathbb R^2$? ...
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Regarding functions from R² to R: continuity and differentiability

Let $f : U \rightarrow \mathbb{R}$ where $U \subseteq \mathbb{R}^2$ is an open set and $P \in U$. I am almost sure the following statements are correct, but please confirm: The only requirement for ...
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2answers
494 views

Real Analysis: Prove that there exists some x ∈ [0,1] such that f(x)=x

If $f\colon [0,1] \to [0,1]$ is a continuous function on $[0,1]$, how can I show that there exists some $x \in [0,1]$ such that $f(x)=x$? I know it will require the Intermediate Value Theorem. ...
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0answers
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Continuity of Joint and Marginal Distributions

If X and Y are jointly continuous then they are individually continuous. Is the converse true?
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2answers
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Find values $f(0)=p$ to make $f(x)=e^{\frac{-1}{|x|}}$ continuous at $x=0$

Given a function $f(x)=e^{\frac{-1}{|x|}}$ when $x \neq 0$ and $f(x)=p$ if $x=0$. I need to find the values for $p$, such that the function becomes continuous at $x=0$. In order to be continuous at ...
3
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2answers
29 views

Continuity of a function through adherence of subsets

We say two sets $A,B$ being $adherents$ if we have $(\overline{A} \cap B)\cup(A\cap\overline{B})\neq \emptyset $. Prove that a function $f:X\to Y$, with $X,Y$ topological spaces, is continuous if ...
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3answers
38 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...