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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An example of a bounded, continuous function on $(0,1)$ that is not uniformly continuous

I can not find the example of a continuous function on $(0,1)$ that is bounded on $(0,1)$, but not uniformly continuous on $(0,1)$. is there any? thank you.
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Let $ f:I \to \mathbb{R} , I=(0,1) $ be uniformly continuous. Then exists $ \lim_{n\to\infty} f(\frac{1}{n}) $

True. Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume: $$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$ Since $ \lim_{n\to\infty} \frac{1}{n} $ is ...
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1answer
26 views

Differentiability implies continuity - A question about the proof

I have a question, to aid my understanding, about the proof that differentiabiility implies continutity. Differentiability Definition When we say a function is differentiable at $x_0$, we mean that ...
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1answer
23 views

Uniform convergence implies continuity and differentiability?

For example: Suppose I have the following series: $$\sum_{k=0}^{\infty}e^{-k}\sin(kt)$$ The Weierstrass-M-Test shows that the series is uniformly convergent on $\mathbb R$. Does this imply ...
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27 views

Continuity and Uniform Continuity on half closed intervals

I have been stuck on the following problem for a long time : Prove that if a function $f:(a,b]\to\mathbb R$ is continuous, then it is uniformly continuous if and only if $\lim_{x\to a^+}f(x)$ ...
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44 views

Uniformly continuous functions on the interval [duplicate]

Let $f:[1,\infty)\to\mathbb R$ be uniformly continuous. Prove $\exists$ $M > 0$ s.t $$\frac{\big|f(x)\big|}{x} \leq M, \hspace{11pt} \forall x\in[1,\infty)$$
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25 views

Examine the continuity of a function $f(x)=\lim\limits_{n\to\infty}(x \arctan(n \cot(x)))$

If we know domain of function $\arctan(x),D_{1}=\mathbb{R}$ and $\cot(x),D_{2}=\mathbb{R}$ without $\{k\pi\}$ we need to check two cases: $x<0$ and $x>0$ How to evaluate limits in these cases? ...
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33 views

About functions and little calculus

Many a times I come upon an $x$ vs. $t$ graph in which the distance $x$ is given as a function of time like $x=f(t)=20+5t^2$. Can its reverse be found? For example, given ...
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32 views

The set of all limits of the image of a divergent sequence under a continuous function

Let $f:\mathbb R\to \mathbb R$ be a continuous function and let $A=\{y=\lim\limits_{n\to \infty}f(x_n):$ for some sequence $x_n\to \infty\}$. My intuition says that $A$ must be a singleton. But I have ...
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1answer
21 views

About the Heine-Cantor theorem.

I don't understand the Heine-Cantor theorem because of one example: The function $x\to \frac{1}{x}$ is not uniform continuous, and we can clearly see in the graph just by looking at the interval ...
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30 views

The functional is continuous

Show that the functional $J(y)=\int_a^b (\sin^3 x+y^2) dx$ is continuous in respect to the $||\cdot||_{\infty}$ norm, at any $y_0 \in C([a,b])$. Let $y_0 \in C([a,b])$. Then for $y \in C([a,b])$ we ...
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1answer
24 views

Closed graph theorem: How do domain and codomain affect continuity?

I had to examine the closed graph theorem under the following circumstances: $X, Y$ metric spaces with $Y$ compact. Does the theorem also hold if Y is not compact? (Assuming compactness in the ...
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2answers
32 views

Continuity and Differentiability of f(x)

$$f(x) = \begin{cases} x^2 + 3x + 2 & \quad \text{if } x \leq 0\\ x^2 - 3x + 2 & \quad \text{if } x > 0\\ \end{cases} $$ Prove that f is continuous at $x = 0$ and not ...
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31 views

Show that for any numbers $p$ and $q$, $\{f\in C[a,b]:p\leq f(x)\leq q\}$ where $x\in [a,b]$ is a closed subset of $C[a,b]$. Similarly for $L_2[a,b]$.

Show that for any numbers $p$ and $q$, $\{f \in C[a,b] \mid \forall x\in [a,b]: p\leq f(x)\leq q\}$ is a closed subset of $C[a,b]$. Similarly for $L_2[a,b]$. We must show that if $f_n\to F$ and ...
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1answer
31 views

f continuous and differentiable?

Consider the function $$f:\mathbb{R}^2\to\mathbb{R}\; (x,y)\mapsto \begin{cases} \frac{x^ay^b}{(x^2+y^2)^c}, & (x,y)\not=(0,0)\text{,}\\ 0, & \text{else } \end{cases}$$ I am trying to ...
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1answer
13 views

Relation between roots of a function and roots of its derivative, IVP

I am troubled with this question of my book: I do know that f (a) = f '(a) = 0 if the multiplicity of root 'a' is greater than 2 but how that fact is exploited here or is there something more ...
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1answer
71 views

How do I analyze/determine the continuity of a function?

My question is really:the following: In general, how do I analyze/determine the continuity of a function? Is there some sort of algorithm? Failing that, here's an example. $$ f: \left]-1,1\right[ ...
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45 views

The functional is not continuous in respect to the strong norm

Let $V=C^1([a,b])$. If $J$ is a continuous functional for the norm $\|\cdot\|_\infty$ then it is continuous for the norm $\|\cdot\|_1:= ||y||_{\infty}+||y'||_{\infty}, y \in V$. But the converse is ...
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1answer
14 views

Distance attained by a function

Let $A$ be a subset of $\mathbb R^n$ and let $x\in \mathbb R^n$. Then $\exists y_0\in A$ such that $d(x,y_0)=d(x,A)$ if $A$ is a non-empty subset of $\mathbb R^n$. $A$ is a non-empty closed subset ...
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45 views

In order to show that a function is C^1 is it enough to show that the 1. partial derivatives exists?

Hello I am having some issues with the following exercise: Let $\textbf{h}: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $\large \textbf{h}(u,v) = u^2 + (v-1)^2 - 5 + e^{u-2}$ (i) Show that ...
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16 views

A Continuous Choice of $k$-Subspaces of a Vector Space Gives a Continuous Choice of Bases

$\newcommand{\R}{\mathbf R}$ The Grassmannian $G_k(\R^n)$ as a topoplogical space is defined in the following way: Let $F_k(\R^n)$ be the collection of all the linearly independent lists of size $k$ ...
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If $f$ is continuous on $[a,b]$ then $1/f$ is bounded on $[a,b].$

$f(x) > 0$ is given for all $x\in [a,b]$. I only got to this: Let $c$ belong to $[a,b]$. Then, for all $ε>0$, there exists $δ>0$, such that, $|x-c|<δ\implies|f(x)-f(c)|<ε$.
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Continuity of the maximum of finite continuous functions

Let $(X,\tau)$ be a topological space and let $f_1,\ldots,f_n:X\to\mathbb{R}$ be continuous functions (the topology of $\mathbb{R}$ is the usual one). Define $g:X\to\mathbb{R}$ by ...
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37 views

Two problems related to continuity of a metric from Munkres' topology book

Let $X$ be a metric space with metric $d$. Show that $d:X\times X\to \mathbb{R}$ is continuous. Let $X^\prime$ denote a space with the same underlying set as $X$. Show that if $d:X^\prime\times ...
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Prove that $f_n(x)$ is discontinuous at $x = 0$.

I am having problems with the following exercise, I am not sure if my procedure is correct. Exercise: Let $ \large f_n(x)=\left\{ \begin{array}{ll} 0 ~~~if~~x = 0 ...
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24 views

Are continuous functions dense in $L^1$?

It is a well known fact that the continuous compactly supported functions are dense in $L^1(\mathbb R)$. An immediate counterexample to this fact for a non locally compact space is $\mathbb R ...
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Absolute continuity and sample paths of Brownian motion

An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of ...
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Continuously extending a set of independent vectors to a basis.

Question: Let $I=(a,b)$ be an interval and let $$v_i:I\to\mathbb{R}^n,\quad i=1,\ldots,k$$ be continuous curves such that $v_1(t),\ldots,v_k(t)$ are linearly independent in $\mathbb{R}^n$ for ...
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$f\circ g$ continuous, $f$ local homeomorphism, $g$ continuous in a different topology $\implies g$ is continuous

I've asked this question before but neglected some assumptions and got a less than useful answer as a result, so I'm going to try again. Let $g:I\times I\to Y$ (where $I=[0,1]$) be a function such ...
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55 views

Problem of Real analysis, continuous functions.

Problem: Let $f$: $\mathbb{R} \to \mathbb{R}$, growing funtion and $D(f)=\{t \in \mathbb{R} : f $ is not continuous in $t \}$. Show that: a) Exist $q: D(f) \to \mathbb{Q}$ such that for all $t \in ...
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If $f(x)$ is continuous at $a$ and $g(x)$ is not continuous at $a$, then can $(f+g)(x)$ be continuous at $a$?

I know that if both $f(x)$ and $g(x)$ are continuous at $a$, then $(f+g)(x)$ would be continuous at $a$. My first thought here is that $(f+g)(x)$ cannot be continuous at $a$ if $g(x)$ is not ...
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Prove that the trigonometric function is uniformly continuous

In my assignment I have to prove that the following function is uniformly continuous in $(0,\frac{\pi}{2})$: $$f(x)=\frac {1-\sin x}{\cos x}$$ Here is my suggestion for solution. Please let ...
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53 views

Proving a statement about a continuous function for which $\forall x\in\mathbb{R},\exists y>x : f(y)>f(x)$

Suppose $f$ is a function which is continuous on $\mathbb{R}$. Also, for all $x\in \mathbb{R}$, there exists $y>x$ such that $f(y)>f(x)$. I must prove that if $\lim_{x\to\infty} f(x)=L$ then ...
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39 views

Integral of $au^2$ where $a$ is continuous and $u \in W_0^{1,2}(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement. Let $a \in C(\Omega)$ and let $u \in W_0^{1,2}(\Omega)$. Suppose that $a > 0$ in $\Omega$ and $\displaystyle ...
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60 views

Fundamental Theorem of Calculus application for $f(x)\geq 0$

Can anybody help me with how to solve the following question using the fundamental theorem of calculus? I'm a bit confused... If $f$ is a continuous function on $[a, b]$ and $f(x)\geq 0$ for all ...
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1answer
33 views

Continuity in closed sets

Please help me, I have being trying this for days now. Let $f:F \to \mathbb{R}$ be a function on a closed set $F$. Show that $f$ is continuous if and only if $A=\{x \in F; f (x) \leq c\}$ and $B=\{x ...
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61 views

Non-continuous topology?

I've been studying topology this term and it really got me interested. But sometimes in math I feel that we are just taught things one by one, without really talking about why we do it that way. So I ...
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Trying to prove that a function got no limit at $(0,0)$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$, defined by: $$ f(x,y)=\begin{cases} 1 & y=x^{2}\\ 0 & \text{otherwise} \end{cases} $$ How can I show that this function got no limit at $(0,0)$? ...
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The derivative of $x!$ and its continuity

is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain. Our teacher said the factorial of a fraction is the fraction itself. He also ...
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20 views

Proving a corollary of a corollary of the Mean Value Theorem (corollary-ception)

This is will a wordy question but here it goes: My analysis book states the mean-value theorem and then a corollary which we will label as (1): Let $f$ be a differentiable function on $(a,b)$ such ...
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How to evaluate limits

Let $f$ be a continuously differentiable function on $\mathbb R$. Suppose that $L=\lim\limits_{x\to \infty}(f(x)+f^{'}(x))$ exists. If $0<L<\infty$, and if $\lim\limits_{x\to \infty} f^{'}(x)$ ...
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43 views

Is $t\mapsto 1_{[0,t]}(s)$ for a fixed $s\ge 0$ continous?

Let $s\ge 0$ and $$f:[0,\infty)\to\left\{0,1\right\}\;,\;\;\;t\mapsto 1_{[0,t]}(s)$$ Is $f$ continuous at $t_0\ge 0$? If $s>t_0$, then $f(t_0)=0=\displaystyle\lim_{n\to\infty}f(t_n)$ for all ...
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Prove that an increasing and surjective function is continuous.

If $f:[a,b]\rightarrow [f(a),f(b)]$ is increasing and surjective, prove that it is continuous. Fix $c \in (a,b)$. Take $\epsilon >0$. We then wish to find the set of $x$ such that ...
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Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist.

I would like to ask you a question about the following question. Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that $\lim_{x \ \rightarrow ...
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1answer
39 views

Using lipschitz estimate to show $|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq |b-a|\sup_{y \in (a,b)}|f'_n(y)-f_p'(y)|$

Assume $(f_n)$ is a sequence of functions that are continuous on $[a,b]$ and differentiable on $(a,b)$. Then using Lipschitz estimate to prove that $$|f_n(x) - f_p(x) - (f_n(c)-f_p(c))| \leq ...
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1answer
23 views

Extending a function continuously from a subset to the whole set

We are given two sets $E$ and $F$ such that $F \subset E \subset \mathbb{R}$. We are given a continuous function $f$ defined on $F$. Can we always extend it to a continuous function on E (not ...
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1answer
62 views

Show $\sum e^{-nx + \cos(nx)}$ is defined on $(a, \infty) $ for any $a>0 \dots$

I want to prove that $\sum e^{-nx + \cos(nx)}$ is defined and continuous on the given interval of $(a, \infty)$ where $a >0$. Then, how exactly do I show it is defined? It just seems trivial, ...
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1answer
32 views

Simultaneous density function of two continuous variables, X and Y.

I'm having issues with calculating the simultaneous density function of two continuous variables, X and Y. I took a screenshot of the information: How should I start? I know that if the two ...
2
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1answer
57 views

How to Find the pointwise limit of $(f_n)$

For $x \in [0, \pi/2]$, if $$f_n(x) = \frac {nx} {1+n\sin(x)}$$ how do you find the pointwise limit of $(f_n)$ ?
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20 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...