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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2
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32 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
2
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3answers
478 views

sequentially continuous on a non first-countable

Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this ...
2
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0answers
82 views
+100

How much does convolution with a C^m kernel increase the order of continuity.

EDIT: I rewrote the question, see the edit history for my previous attempt. My questions are: Is the following proof/reasoning essentially correct? How do I make it more precise, concise, correct? ...
3
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3answers
44 views

Why does $\lim_{x\to 0} \frac {\sin (xy)}{x} \to y $?

Let $f(x,y) = \frac{\sin (xy)}{x}$ for $x\neq 0$. How should you define $f(0,y)$ for $y\in \mathbb{R}$ so as to make $f$ a continuous function on all of $\mathbb{R}^2$? So in order for a function to ...
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0answers
19 views

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continous on $(0,\pi)$

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continuous on $(0,\pi)$ I think about showing the uniform convergence of $$ f_k: \mathbb (0, \pi) ...
0
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0answers
21 views

Question concerning continuity of composite functions

Consider two functions, $a(r)$ and $b(r)$. If a is continuous at $c$,and $b$ is continuous at $a(c)$ , then $b(a(c))$ is continuous at $c$ .(This is a theorem stated in the text Thomas' Calculus) Now ...
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2answers
226 views

Does intermedia value theorem apply to continuity on open intervals?

Does the intermediate value theorem apply to functions that are continuous on the open intervals (a,b)? I know it's pretty vital for the theorem to be able to show the values of f(a) and f(b), but ...
2
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0answers
48 views

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces.

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces. I have seen some example which uses $X$ to be non ...
1
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3answers
44 views

Show that the function is continuous

To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}$ with $f=\left\{\begin{matrix} \frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\ 0 & , (x,y)=(0,0) \end{matrix}\right.$ is ...
0
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3answers
48 views

Intuitive meaning of the probability density function at a point

I understand how to integrate probability density functions to find probability within a certain range. However, what I don't understand is what it would mean to set the variable (say x or y) to a ...
2
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1answer
31 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
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1answer
42 views

Is this function bounded or not?

$f(x) = \left(1-\frac ax\right)^2$ where both $x>0$, $a>0$ Is this function bounded? i.e. is there an M such that $f(x) ≤ M < \infty$ ? How can I figure this out? Thanks very much in ...
1
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2answers
55 views

Explain why continuity along straight lines is not enough to conclude continuity

Consider the function with domain $A = \{ (x,y) \in \, \mathbb{R}^2: (x,y) \neq (0,0)\}$ given by $$\frac{2x^2y}{x^4+y^2}$$ Letting $(x,y)$ approach $(0,0)$ along the straight line $y=ax$ , where ...
0
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1answer
18 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
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0answers
34 views

Attaining a maximum without applying the Weierstrauss Theorem

The example that mookid gave in this question is a good one. There is no maximum since it is not continuous. How could you explain why the function given by mookid will attain a maximum on any compact ...
1
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2answers
115 views

Unbounded function on compact interval?

So what are some unbounded function on compact interval, if there is any? Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?
0
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1answer
29 views

A function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.

I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous. I've seen from a ...
0
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2answers
52 views

Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.

Let $C^1([0 , 1])$ be the subspace of $C([0 , 1])$ consisting of the functions that have a continuous derivative throughout $[0 , 1]$. Show that the mapping $\Psi:f → f~'$ from $C^1([0 , 1])$ to $C([0 ...
2
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1answer
35 views

$f:[a,b]\to \mathbb R$ is continuous , has a finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?

If $f:[a,b]\to \mathbb R$ is a continuous function having finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?
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1answer
39 views

$f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ?

If $f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ? Please help . Thanks in advacne
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0answers
47 views

about cauchy sequence in metric space [on hold]

Let $f$ be a function from a metric space $(X,d_1)$ to a metric space $(Y,d_2)$. If the image of every Cauchy sequence in $X$ is a Cauchy sequence in $Y$, how can I prove that $f$ is continuous?
2
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1answer
67 views

Value of $x \log x$ at $x=0$ [duplicate]

What is the value of $f(x) = x \ln (x)$ at $x= 0$? Is it $0$ or indeterminate?
3
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0answers
52 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
0
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0answers
27 views

Continuity of Function involving logarithm function

I want to prove a function $f(x) = g(x) * log x $ is continuous on interval $[0, 1]$, where value of $g (x)$ is $0$ at lower limit point $0$. Anybody can help me out here. Thanks in advance.
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0answers
31 views

Functional Analysis: Continuity of operators [on hold]

Which of the following operators are continuous: a) $A:L^2 [0,1]\rightarrow L^2 [0,1]$ defined by the formula $\displaystyle (Ax)(t)=\int \limits_{0}^{1} K(t,s) x(s)ds$, where $K(t,s) \in L^2( ...
1
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1answer
334 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
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1answer
14 views

How to check F:AxI->B is continuous

A and B are topological spaces.Let f and f' are continuous maps from A to B and homotopic.Then we need F:AxI->B,continuous,where F(s,0)=f(s) and F(s,1)=f'(s). Now my question is if we want to ...
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1answer
20 views

Average integral for continuous functions with compact support

Let $f$ be a continuous function with compact support in $\mathbb{R}^n$. Show that \begin{equation} \lim_{r\to 0} \frac{1}{|B_r(x)|} \int_{B_r(x)} f(y)\,dy = f(x), \end{equation} where $B_r(x)$ is the ...
2
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3answers
44 views

When the set of $r$-far interior points from a set is open

Let $E$ be a subset of a metric space $X$ and for $r > 0$ let $$ E_r = \lbrace x \in E : d(x,E^c) > r \rbrace .$$ Is the set $E_r$ always open? Equivalently, is the function $ x \mapsto ...
2
votes
3answers
154 views

Can the Intermediate Value Theorem be proved by Heine-Borel Lemma?

Can the Intermediate Value Theorem be proved by Heine-Borel Lemma, and how? I mean "Every open cover of close interval has a finite subcover", without compactness etc. Because in class we proved it ...
1
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1answer
11 views

Proof that a derivative's points of discontinuity are all essential

I'm reading Wikipedia's article on Darboux's theorem, and it says the following: "Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the ...
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1answer
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2
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5answers
136 views

A-noncompact, Does there **always** exist a continuous function $f: A \to \mathbb R$ which is bounded but does not assume extreme values?

It's well known that if $ A \subset \mathbb R$ is compact then every continuous function $f:A \to \mathbb R$ is bounded and assume extreme values .So the obvious question is: Given any non compact ...
2
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1answer
57 views

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.

Prove that the set of square matrices $A(x)=\begin{pmatrix} 2x+y & x \\ 3x & 2x+3y \\ \end{pmatrix}$ for $x,y\in [0,1]$ is a compact set.(Take into consideration metric $d_2...$) I was ...
1
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1answer
30 views

Let $([0,1],\mathcal{B}([0,1]),\lambda)$, $\lambda$ Lebesgue measure in $[0,1]$.

Show that if $f$ is $p$-integrable then, for each $\epsilon>0$, exists a function $h$ which is continuous in $[0,1]$ s.t. $\|f-h\|_p\leq\epsilon$. Is there any simpler way to show it than ...
0
votes
1answer
28 views

Is there proof anywhere of the continuity of spherical coordinates and cylindrical coordinates?

Im thinking they are continuous as a composition of continuous functions, but then again. I don't know exactly which specific(precisely speeking) functions are in question.. Any thoughts on this?
4
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1answer
23 views

How would I make continuous functions to form these sets? Parametarizing of sets

How would I make continuous functions to form these sets?(So the domain is connected) I need continuous functions that map connected sets to these in question. $1. \text{Cone}$ $$(x,y,z)| \ ...
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2answers
27 views

continuous map of connected set is connected, example: Proving the connectedness of this set.

I thought I would try to use this to prove connectedness in this set if possible: $$\{(x,y)\mid 1<x^2+y^2<4\}$$ $f(x,y)=x^2+y^2$ So since $(1,4)$ is connected in $\mathbb R$ so it this set, as ...
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3answers
57 views

To prove continuity using sequential definition of continuity

I have to show that the function $$f(x) = \begin{cases}\;\; x &, \text{ if } x \text{ is rational} \\ - x &, \text{ if } x \text{ is irrational} \end{cases}$$ is continuous at $0$ and ...
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57 views

How do i prove that $f(x) = \sin\frac{1}{x}$ is continous for all $x \in \mathbb{R}$ except $0$. [closed]

How do I prove that $$f(x) = \sin\frac{1}{x}$$ is continuous for all $x \in \mathbb{R}$ except $0$. At $0$ I can show it is discontinous, but how to show its continuity at other points
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1answer
24 views

Continuous function not sobolev

Let $I=(a,b)$ an open bounded interval. It is well known that $W^{1,p}(I)\subset C(I)$. It easy to see that there are $f\in C(I)$ such that $f\notin W^{1,p}(I)$ It is enough to take $I=(0,1)$ and ...
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2answers
40 views

Given any non compact set $A \subset \mathbb R^n$ does there exist a continuous function $f: A \to \mathbb R$ which is not uniformly continuous?

It's well known that if $ A \subset \mathbb R^n$ is compact then every continuous function $f:A \to \mathbb R$ is uniformly continuous.So the obvious question is: Given a non compact set $A \subset ...
1
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1answer
21 views

Lusin property (N) for functions of several variables

I just read in a paper by Martio and Zeimer$^1$ that smooth functions ($C^1$) of several real variables have the have the Lusin property (N). I have two questions. First, could someone give me a ...
2
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1answer
632 views

Show that the upper envelope of a bounded function is upper semi continuous directly

Definition 1: A real valued function $f$ is said to be upper semicontinuous at a point $p$ if: $$f(p) \geq \limsup_{x \rightarrow p} f(x) $$ Definition 2: Let $f$ be a bounded real valued ...
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1answer
45 views

Two variable function that's continuous on all linear paths, but nevertheless discontinuous

Suppose we want to know $\lim_{(x,y)\rightarrow (0,0)}{f(x,y)}$. The epsilon-delta definition of continuity (in $\mathbb{R}^n$) implies that "all paths" to $(0, 0)$ must result in the same limit for ...
2
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1answer
27 views

Uniformly $\beta$-continuous functions (jumps no greater than $\beta$) converge uniformly to $f$, is $f$ continuous?

Let $(X,\rho)$ be a compact metric space, we say a function on $X$ is uniformly $\beta$-continuous if, for every $\epsilon > 0$, there exists $\delta > 0$ such that if $\rho(x,y) < \delta$, ...
2
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1answer
44 views

Integral should not be continuous.

I'm looking for a counter-example: Let $f:[0,1]\times \mathbb R\to\mathbb R$ be continuous in such a way that $$F(x):=\int_{\mathbb R} f(x,t) dt$$ defines a function $F:[0,1]\to\mathbb R$ (so in ...
4
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1answer
67 views

On proving the continuity of $\arctan$

I read that continuity of $\arctan$ follows from the equation $\arctan x + \arctan 1/x = \pi / 2$ ($x> 0$) and $\arctan x + \arctan 1/x = -\pi / 2$ ($x<0$). But I can't work out how this ...
2
votes
1answer
475 views

Continuity and differentiability of function series

Examine the continuity and differentiability of functions: a) $\displaystyle f(x)=\sum_{n=1}^{+\infty}\frac{\sin(nx)}{n^3}$ b) $\displaystyle ...
3
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1answer
68 views

Prove the Lipschitz constant must be less than 1.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...