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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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
48 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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52 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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2answers
40 views

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
65 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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1answer
28 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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1answer
67 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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1answer
29 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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1answer
33 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
21 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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1answer
36 views

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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1answer
55 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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1answer
38 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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0answers
22 views

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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1answer
35 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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2answers
53 views

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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1answer
223 views

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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1answer
111 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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1answer
24 views

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
63 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
63 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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0answers
19 views

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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18 views

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
49 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
40 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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0answers
46 views

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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1answer
20 views

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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1answer
103 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
50 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
40 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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2answers
77 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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2answers
33 views

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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32 views

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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0answers
13 views

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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1answer
25 views

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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2answers
39 views

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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0answers
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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1answer
67 views

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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0answers
158 views

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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4answers
503 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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1answer
54 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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2answers
39 views

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
24 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
65 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
45 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
44 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 ...
2
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1answer
36 views

$\varepsilon$-$\delta$ proof of continuity of floor function $\lfloor x\rfloor$

I would just like to ask someone to confirm or correct the following 'proof' of continuity of the floor function. Let $\varepsilon>0$ be given. Set $\delta:=\min\lbrace x-\lfloor x\rfloor,\lceil ...
0
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1answer
14 views

continuity of a function f = (f_1,f_2) in a product topology if f_1 and f_2 are continous

Say $X$, $Y_1$ and $Y_2$ are topological spaces. Let $f_1 \; X \to Y_1$ and $f_2 \; X \to Y_2$. If $f\; X \to Y_1 \times Y_2 $ $f(x) = (f_1(x), f_2(x))$ $Y_1 \times Y_2$ is a topological space with ...
1
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2answers
59 views

Find $\alpha$ and $\beta$ so that $f(x)$ is continuously differentiable

The function $f(x)$ is defined as following $$ f(x) := \begin{cases} \cos x+e^x, & \text{if $x < 0$} \\ \ \alpha(1+x)^{2009}+\beta e^{-x}, & \text{if $x \ge 0$} \end{cases} $$ I need to ...
1
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1answer
33 views

Explaining the one-dimensional continuity equation with respect to density evolution

I've got a rather abstract question So the continuity equation for a one-dimensional continuum is: $$ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho v)=0 $$ and we can expand ...