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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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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37 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
25 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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1answer
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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30 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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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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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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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
15 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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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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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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15 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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46 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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1answer
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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42 views

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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1answer
37 views

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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1answer
38 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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0answers
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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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1answer
25 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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1answer
23 views

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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1answer
56 views

$f\circ g$ 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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1answer
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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3answers
46 views

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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1answer
54 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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1answer
41 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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6answers
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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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1answer
64 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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1answer
35 views

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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3answers
61 views

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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1answer
22 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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2answers
29 views

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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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 ...
3
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3answers
60 views

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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2answers
66 views

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 ...
4
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1answer
63 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 ...
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1answer
59 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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1answer
24 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 ...
3
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2answers
59 views

Proving that a function is discontinuous

In my assignment I have to prove that the following function is discontinuous: $$f(x)=\begin{cases}2x-1&\text{if }x\notin\Bbb Q\\x^2&\text{if }x \in \Bbb Q\end{cases}$$ I have to prove that ...
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22 views

A question involving continuity with respect to the product topology

Let H be a nonempty set, $\cdot$ a binary operation on H, $\Gamma$ a topology on H and $$\varphi : H \times H \to H, \;\; \varphi(x, y) = x y, \;\; \forall x, y \in H$$ continuous with respect to the ...
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2answers
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Sequential Definition of continuity || Modulus Property

I am stuck up with these questions from my text book on sequential continuity : {My questions might sound trivial a bit trivial} I am not able to figure how its being written that $|f(X_n)| \leq ...
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1answer
33 views

Is the function $f(x)=x^2$ absolutely continuous on the real line?

In Wiki (Lipschitz), it says: A Lipschitz function $g : \mathbb{R}\to \mathbb{R}$ is absolutely continuous. According to the definition of absolute continuity, I am confused about an simple ...
0
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2answers
28 views

Continuous and bounded - Check my proof please

Let $f : [0, ∞) → \mathbb{R}$ be continuous such that $\lim_{x→+∞} f(x) = 0$. Prove that $f$ is bounded on $[0, ∞)$ By our hypothesis and the definition of continuity, given $ c \in [0, \infty), ...
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1answer
39 views

Prove that $f$ in monotonic

In my assignment I have to prove the following: Let $f$ a continuous function in $\Bbb R$. Prove the following: if $|f|$ is monotonic increasing, in R then $f$ is monotonic in R. ...
0
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2answers
33 views

Solving $f(x) = x^5 +x + 1 = 0$ with halving the interval / bisection method

Question: Use halving the interval / bisection method to approximately solve: $$f(x) = x^5+ x + 1 = 0$$ with a precision of $\pm 0.1$ Attempted solution: The general idea, as I understand it, is ...