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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66 views

Prove that f is discontinuous at all the real points except 0 and 1

My problem is as follows- Let $f : \mathbb R \to \mathbb R$ be defined in the following manner $$f(x) = \begin{cases} x & \text{if $x$ is rational,} \\x^2 & \text{if $x$ is irrational.} ...
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1answer
41 views

Weakest topology equivalence

Prove the equivalence of the following $Y \subset X$ has the subspace topology . $Y$ has the weakest topology to make the inclusion $i:Y\to X$ continuous. For all topological spaces $Z$ and maps ...
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1answer
9 views

Proving Continuity & Adding Discontinuous Functions

I've been wondering, how do you exactly prove that a function is continuous everywhere (or within the domain in which the function is defined)? Given some curve, my current approach would be to to try ...
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1answer
23 views

What is meant by the continuity of the Hessian matrix

I have a simple and short question: "What is meant by the continuity of the Hessian matrix?" I guess it means that all the second partial derivatives of a function $f$ are continuous functions? is ...
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1answer
37 views

Prove if $E$ is a Lebesgue measurable set, there exists a continuous function $f$ differing from $\chi_{E}$ on a set of measure $< \epsilon$?

I am reviewing my analysis notes, and I don't really understand the proof given by my professor. He first proved if $E$ is a Lebesgue measurable set and $\epsilon > 0$, then there is an open set ...
2
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1answer
19 views

Uniform continuity problem

Let $f(x)$ be continuous on $[0, \infty)$, $f'(x)$ and $f''(x)$ be continuous on $(0, \infty)$. Which of the following statements are true: I. If $f'(x) > 0$ and $f''(x) < 0$, then f(x) is ...
7
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2answers
140 views

$f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant.

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that $$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.
2
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1answer
68 views

Equicontinuity of a pointwise convergent sequence of monotone functions with continuous limit

I was looking at this question, and trying to come up with a counterexample. After thinking about it, I thought the following might be true: Claim: let $\{f_n\}$ be a sequence of continuous, ...
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1answer
53 views

Find $k$ so that $f(x)$ is a continuous function [closed]

Find $k$ so that $f(x)$ is a continuous function. $$f(x)=\left\{\begin{array}{ll}x^2 &x\leq2\\ k-x^2 & x>2 \end{array}\right.$$ Does anyone know how to go about this problem? Thanks in ...
2
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2answers
37 views

If $f^{n_o}$ has a fixed point , then does $f$ also has a fixed point , where $f$ is continuous on $\mathbb R$?

In relation to this question , To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$ , if $f: \mathbb R \to \mathbb R $ is a continuous function such that for some $n_o \in \mathbb N$ the ...
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0answers
24 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
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2answers
54 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, wether if this topology ...
2
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1answer
40 views

Equivalence of continuous functions

Consider two topological spaces, $X$ and $Y$, and two continuous functions $f$ and $g$. By definition, given an open set $S$ in $Y$, the pre-image of $S$ under $f$ (or $g$) is an open set of $X$. Let ...
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3answers
78 views

How to prove, that $e^x$ is uniformly continuous if $x$ is negative?

How can one show with only elementary mathematics, that $e^x$ is a uniformly continuous function on $(-\infty;0]$ I started with $\mid e^x-e^y \mid$, knowing that I assume , that $\mid x-y ...
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1answer
45 views

How to show that a complex-valued function is uniformly continuous?

should a function be uniformly continuous in both arguments if it should be uniformly continuous as a complex-valued function. For example how can I proove that ...
1
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1answer
26 views

Lipschitz continuity for generalized inverse matrix

Suppose $A$ and $B$ are full-rank and well-conditioned. Is Lipschitz continuity held for generalized inverse? $$\|A^+ - B^+\| \le \omega \|A-B\|,$$ for some $\omega > 0$, where the norm could ...
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2answers
86 views

on Continuous and Open Functions

Let $X,Y$ be compact Hausdorff spaces. If $f$ is a continuous function from $X$ onto $Y$, then $f$ is open. I am asking can the above result be proved. I am aware of the following cases: If $f$ ...
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1answer
71 views

Uniform Continuity $\implies$ Continuity

In metric spaces it is a well known fact that uniformly continuous functions are indeed continuous at any point. What about uniform spaces? How can I prove this? (with the definition of topology in ...
6
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1answer
79 views

Topology/continuous functions of $\mathbb{R}^n$ using paths.

We know that for a function $f:\mathbb{R}^n\to\mathbb{R}$ to be continuous, it is not sufficient for itto be continuous with respect to each coordinate. I believethe most commom counter-example is the ...
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0answers
25 views

Another question on continuous surjective functions

It is known thta if $f: \mathbb R \to \mathbb R$ is a continuous surjective function that takes every value at most twice then $f$ is strictly monotone . My question is " What is the maximum value of ...
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2answers
35 views

On continuous surjective function that takes every value at most a finite no. of times

If $f: \mathbb R \to \mathbb R$ is a continuous surjective function which takes every value at most a finite number of times , then is it true that $f$ is strictly monotone ?
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1answer
19 views

Continuity of function does not imply continuity of extension

Let $f$ be increasing on a dense subset $D$ of $\mathbb{R}$, and define $\tilde{f}$ on $x\in\mathbb{R}$ $\tilde{f}(x):=\inf_{x<t\in D}f(t)$. Show that the continuity of $f$ on $D$ does not imply ...
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1answer
57 views

Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...
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1answer
25 views

Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
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0answers
30 views

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 ...
2
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1answer
21 views

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 $x \in [c,d]$ ...
1
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2answers
26 views

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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0answers
21 views

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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1answer
40 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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0answers
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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2answers
39 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
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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1answer
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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1answer
57 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
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 ...
3
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1answer
29 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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0answers
22 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
44 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
31 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
21 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
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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2answers
49 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 ...
5
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1answer
217 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
109 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
59 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: ...
1
vote
3answers
56 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 ...
0
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0answers
16 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 ...