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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A function is discontinuous at all rational points and continuous at all irrational points [duplicate]

Define $f(x)$ for $x\in[0,1]$ by $f(\frac pq)=\frac1q$ if $p$ and $q$ are relatively prime, and $f(x)=0$ if $x$ is irrational. How can we see that $f$ is discontinuous at all rational points and ...
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if $f([a,b])=[c,d]$ and $[c,d] \subset [a,b]$, is there $x \in [c,d]$ such that $f(x)=x$?

I'm trying to prove something that I'm not sure is correct. Let $f$ be a continuous, differentiable and monotonic function $f:[a,b] \to [c,d]$, where $[c,d] \subset [a,b]$. Is there an $x \in [c,d]$ ...
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If $f: \mathbb{R}^2\to \mathbb{R}$ is Lipschitz, then $g(x)=f(x,a)$ too?

Let $f: \mathbb{R}^2\to \mathbb{R}$ be a Lipschitz continous function, meaning that $ |f(x_1,y_1) - f(x_2,y_2)| \leq L \, || (x_1,y_1) - (x_2,y_2) ||$ for all $(x_i,y_i) \in \mathbb{R}^2$, where ...
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Existence of increasing, smooth modulus of continuity

First, recall the definition: Given a function $f:M\to N$, where $M$ and $N$ are metric spaces, a modulus of continuity for $f$ is a function $\omega:[0,\infty)\to[0,\infty)$ such that ...
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39 views

What am I doing wrong in this continuity check?

I want to show that the function $f$ is discontiunous. $f$ is defined as follows: $$f(x,y) := \begin{cases} (x^2+y^2)\cdot\sin(\frac{1}{\sqrt{x^2+y^2}}) & , (x, y) \neq (0,0) \\ 0 ...
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1answer
42 views

Which function is not uniformly continuous? [closed]

Which of the following functions is not uniformly continuous? $$A.\ \ \ \frac{1}{x}, \ \ \ x \in [1, +\infty)$$ $$B. \ \ \ \ \ \ \ \frac{1}{x}, \ \ \ x \in (1,2)$$ $$C. \ \ \ \ \ \ \ \ \frac{1}{x}, ...
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51 views

How to show $f(x)=\exp((|x|^2-1)^{-1})$ if $|x|<1$ and $f(x)=0$ if $|x|\geq 1$ is a test function?

What would be the formal argument for showing the function $f:\mathbb R^n\longrightarrow \mathbb R$, $$f(x):=\left\{\begin{array}{ccc} ...
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Is $f:\mathbb E^1\to X$ continuous?

$f(x)=x$. $X$ is the set of all real numbers with finite complement topology (A set is open in this space iff it's complement is finite).
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3answers
61 views

If $f$ is continuous, so is $g=|f|$ [closed]

Prove that if $f$ is continuous, so is $g=|f|$. I need help on this. Thank you. Ok, this is my first time here. The definition of continuity i am using is that $f$ is continuous at $a$ if for any ...
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26 views

how to show that $f_n$ is nonnegative on an open interval for all $n$ large enough

Let $\{f_n\}_{n=1}^\infty$ be a sequence of continous functions on $[0,1]$ and for all $x\in [0,1], f_n(x)$ is eventually nonnegative. Show that there is an open interval $I\subseteq[0,1]$ such that ...
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55 views

$f_a(x) = e^{ax}$ is uniformly continuous over $[0, \infty)$?

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f_a(x) = e^{ax}$. a) Prove that $f(x) = e^x$ is not uniformly continuous. b) Determine for wich values of $a$ the function $f_a(x)$ is ...
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23 views

Continuity & boundedness on open interval implies uniform continuity

Suppose f(x) is continuous and bounded on (0,1). Is f(x) uniformly continuous on (0,1)? I think yes, because it's bounded, i.e. there exists $M: |f(x)| < M$. We could use this M as $\delta$ in the ...
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27 views

Show that $Y$ is not path-connected

Let $\mathbb{R}^2$ with the usual topology and let $$ Y = A_0 \cup (\bigcup_{n \in \mathbb{N}} A_n) \cup (\bigcup_{n \in \mathbb{N}}L_n)$$ where $$ A_0 = \{ 0 \} \times [0,1] \qquad A_n = \{ ...
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1answer
20 views

Finding all continuity and differentiability points of a function

Let $$f(x) = \begin{cases} x^2(x^2-1),&x \in\mathbb{Q} \\ 0,&x \not\in\mathbb{Q} \end{cases}$$ A. When is this function continuous? when is it differentiable? I solved these kind of ...
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Lipschitz continuity of inverse

Given a function f : $\mathbb{R}^n\to\mathbb{R}^m$, which is known to be Lipschitz continuous, can we say anything about the Lipschitz continuity of it's inverse function (in this case, the ...
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42 views

Prove that the inverse image of an open set is open

Let $ X \subset \mathbb{R}$ be a non-empty, open set and let $f: X \rightarrow \mathbb{R}$ be a continuous function. Show that the inverse image of an open set is open under f, i.e. show: If $M ...
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27 views

How to find the average value of a discontinuous function

A car covers $\frac{1}{3}$ distance with speed $20\frac{m}{s}$ and $\frac{2}{3}$ with $60\frac{m}{s}$. What is the average speed over the entire interval? Due to the discontinuity of the function, I ...
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Let $f:(0,\infty) \to \mathbb{R}$ s.t f'(x)>x. Prove that f is not uniformly continuous [duplicate]

I'm trying to prove the following statement: Let $f:(0,\infty) \to \mathbb{R}$ s.t f'(x)>x. Prove that f is not uniformly continuous. My first step was thinking about Lagrange, so I wrote that ...
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32 views

Locally Vs Globally Lipschitz Confusion

Is there any difference in a function being locally Lipschitz on $\mathbb{R^n}$ and being globally Lipschitz?
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Proving $ f(x)=(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$

Prove that $f(x)=\Large(\frac {\sin x} {x})^{\frac {1} {x^2} }$ is uniformly continuous on $(0,1]$. Basically what I need to show here is that there is a limit 'from the right' for $x=0$ so the ...
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Topology: Opens vs Neighborhoods

Disclaimer: This thread is meant informative and therefore written in Q&A style. The problems are highlighted in bold face. The axiomatization of topology can be done in various ways all of ...
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108 views

showing $\int _a^b\left(f'\left(x\right)\right)dx\:=\:f\left(b\right)-f\left(a\right)$

Let $f(x):[a,b]\to \mathbb R$, be differentiable on $[a,b]$ (and continuous) so that $f'(x)$ is integrable on $[a,b]$. I need to show that: $$\int _a^b\left(f'\left(x\right)\right)\mathrm dx = ...
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Tomae's Popcorn Function: Preimage of Opens?

I'm just wondering what the preimage of an (open) neighborhood say $(-0.5,0.5)$ containing the point $T(\frac{1}{\sqrt{2}})=0$ under Tomae's popcorn function $T$ looks like. Does somebody have an ...
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1answer
57 views

Why this way of showing that $\sin x$ isn't uniformly continuous is wrong?

I know $\sin x$ is uniformly continuous and it was asked before (Prove $\sin x$ is uniformly continuous on $\mathbb R$). My question is related to this answer: ...
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3answers
55 views

Show $f$ is uniformly continuous

Let $f$ continuous function on $[0,\infty)$. Lets assume there are $a,b$ such that: $\lim_{x\rightarrow \infty} f(x)-(ax+b) = 0$. Prove $f$ is uniformly continuous on $[0,\infty)$. Well, At ...
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Continuity of a function defined by an improper integral.

What is the result that allows us to say the following: In order to show that some function $f(x)= \int_0^{+\infty}g(x,t)dt$ is continuous on $[0,+\infty)$ we show that $g$ is continuous on ...
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Relation between continuity as a map and joint continuity

Let $f=f(x,y) : \mathbb{R}^2 \to \mathbb{R}$ and denote by $C(\mathbb{R})$ the space of bounded and continuous, real-valued functions on $\mathbb{R}$. Is it true that if the map $x\mapsto f(x,\cdot)$ ...
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1answer
33 views

uniform continuity on $(a, b]$ implies limit at $a^+$ exists and finite

Let a uniformly continuous function $f$ on $(a, b]$. Prove that $\lim_{x\rightarrow a^+} f(x)$ exists and finite. What I did so far: from the definition of uniform continuity: ...
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1answer
38 views

The continuous dual of the reals

I just have a few questions involving the continuous dual of $\mathbb{R}^{N}$. We know that the dual $(\mathbb{R}^{N})^{*}$ of $\mathbb{R}^{N}$ is the space of all linear forms $$a: \mathbb{R}^{N} ...
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Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
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semicontinuity implies sequential semicontinuity

I have that $F:X\to (-\infty,+\infty]$, with $X$ topological space. By definition, $F$ is lower semicontinuous in $x_0 \in X$ if $\forall t \in \mathbb{R}: \. t<F(x_0) \.\exists U\in ...
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102 views

$f(x,y) = (1- \cos(\frac{x^2}{y})) \sqrt{x^2+y^2}$

Let $f(x,y) = (1- \cos(\frac{x^2}{y})) \sqrt{x^2+y^2}$ for $y \ne 0$ How can I prove that f is not differentiable in $(0,0)$. Please some help.
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1answer
47 views

Two continuous function differ on set of measure zero?

Is it correct that two continuous functions $f,g: \mathbb{R}^n \to X$, where $X$ is a topological space, cannot differ only on a set of measure zero? So as a consequence, for instance, there is at ...
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1answer
38 views

Is this function continuous on transcendental number

This question is motivated from Thomae's function continuity at irrationals together with the fact that transcendental numbers are dense in real numbers. Let $$f(x) = \begin{cases}1 &, \text{x ...
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73 views

Continuous Map: Open $\iff$ Closed? [closed]

Is it true that a continuous map is open iff it is closed: $$f\text{ continuous}:\quad f\text{ open}\iff f\text{ closed}$$ The idea is that when somebody asks for embeddings, quotient maps and ...
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How to show $\{f_n\}_{n=1}^\infty$ has uniformly convergent subsequence on [0,1]?

Let $\{f_n\}_{n=1}^\infty$ a sequence of second order differentiable functions on the interval [0,1]. If $\forall n\in \Bbb N$ $f_n(0)=f_n'(0)=0$ and for all $n\in \Bbb N$ and $x \in [0,1]$ , ...
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Question about a theorem concerning the continuity of integral functions

If we have $$F(t):=\int_V f(t,x)dx$$ where $V$ is some measurable subset of $\mathbb R$ and $x\mapsto f(t,x)$ is a measurable function. Moreover let $F$ be defined for all $t\in U$ a open subset of ...
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Continuity of the solution to a matrix PDE (mapping of a parameter to solution)

I'm considering the following PDE in $\Phi$: $\frac{\partial \Phi(t,s)}{\partial t}$ + $sR\frac{\partial \Phi(t,s)}{\partial s}$ + $\frac{1}{2} s^2 M \frac{\partial^2 \Phi(t,s)}{\partial s^2}$ + ...
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Inverse of Continuous Function on Closed Bounded Part of R. Why Bounded?

Consider the following proposition: Let $A$ be a closed bounded part of $\Bbb R$. Assume $f: A\rightarrow \Bbb R$ is a continuous injective function. Then $f^{-1}: f(A) \rightarrow A$ is also ...
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Epsilon delta prove for continuïty$ (1-\cos(|xy|))/y^2$

Let a function, $\mathbb{R}^2\to\mathbb{R}: \begin{Bmatrix} \frac{1-\cos(|xy|)}{y^2}&y\neq0\\ \frac{x^2}{2}&y=0 \end{Bmatrix} $ I have to prove this is continious. For y$\neq 0$, this is ...
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45 views

implicitly define a function

The first part i made $u=\frac{z}{x}$ and $v=\frac{y}{x}$ and after calculating the partial derivatives $\frac{dz}{dx}$ and $\frac{dz}{dy}$ The second i have no idea how to do it
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The linearity of $D \beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathcal{L}(\mathbb{E_1} \times \mathbb{E_2},F)$

Let $\mathbb{E_1}, \mathbb{E_2}$ and $\mathbb{F}$ normed spaces of finite dimensions and $\beta : \mathbb{E_1} \times \mathbb{E_2} \rightarrow \mathbb{F}$ is one bilinear function. Then $D \beta : ...
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Differentiability-Related Condition that Implies Continuity

I previously asked a related question here that I did not phrase as I intended. This is a revision of that question: It is a well-known fact that differentiability implies continuity. And, for ...
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490 views

Weaker Condition than Differentiability that Implies Continuity

It is a well-known fact that differentiability implies continuity. My question is this: is there some condition for a function that is both weaker than differentiability and stronger than continuity? ...
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45 views

Real analysis help: Proof of continuous functions

The question is: Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be continuous on $\mathbb{R}$ satisfying $h(m/2^n)=0$ for all $m\in \mathbb{Z},n\in \mathbb{N}$. Show that $h(x)=0$ for all $x\in \mathbb{R}$. ...
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Question about limit and continuity

I have that $u_0>0$ , $u_n=u_n^+-u_n^{\raise{1pt}{-}}$ and $u\mapsto u^{±}$ is continuous if $u_n\rightarrow u_0$ why we have that $u_n^+\rightarrow u_0$ and $u_n^{\raise{1pt}{-}}\rightarrow 0 $ ...
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1answer
21 views

$f(x)$ non-decreasing then pseudoinverse of $x + f(x)$ is Lipschitz.

while studying some proof, I came across the following statement: Let $f$ be a non-decreasing function defined on closed interval $[a, b]$. Let $\alpha = a + f(a)$ and $\beta=b+f(b)$. We can ...
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3answers
64 views

prove that $f(x)=\sum _{n=0}^{\infty}\frac{\cos(nx)}{2^n}$ is continuous

I refered that each fn is continuous because its the fraction of a continuous function by a number and so $f(x)$ that is the sum of continuous functions is continuous. Is it right?
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3answers
44 views

Show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f$ is discontinuous at $c$

How to show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f: \mathbb R \rightarrow \mathbb R$ is discontinuous at $c$ ? I know that $f$ cannot have ...
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1answer
40 views

Are the two statements about continuous functions equivalent?

I have always wondered about this: A continuous function is defined thus: for any $\epsilon>0$, there exists $\delta\in\Bbb{R}$ such that $|x-y|<\delta\implies |f(x)-f(y)|<\epsilon$ for ...