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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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 ...
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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 ...
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1answer
24 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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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 ...
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1answer
17 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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2answers
54 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 ...
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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 ...
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1answer
28 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 ...
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2answers
51 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 ...
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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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1answer
34 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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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?
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0answers
47 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: ...
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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?
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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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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( ...
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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 ...
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3answers
41 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 ...
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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
34 views

Prove or disprove about isomorphic functions [on hold]

Prove that : if f is an isomorphic then it is continuous or not?
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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 ...
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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?
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1answer
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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
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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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How do i prove that $f(x) = \sin\frac{1}{x}$ is continous for all $x \in \mathbb{R}$ except $0$. [on hold]

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
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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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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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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 ...
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2answers
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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 ...
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1answer
29 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 ...
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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 ...
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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 $\rho(x,y) < \delta$, ...
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1answer
43 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 ...
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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 ...
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1answer
16 views

Definition of continuously differentiable for functions of several variables

When we say that a function $f:\mathbb{R}^m\to\mathbb{R}^m$ is $C^1$, what exactly does this mean? Does it mean that all the directional derivatives are continuous individually (I am sure not), or ...
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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 ...
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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 ...
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1answer
39 views

covariance matrix of X+Y and X-Y

This question comes up in almost every past paper i do and is worth 10 marks and just can't work it out... Let $X$ and $Y$ have the joint pdf $$f(x,y)= \begin{cases} e^{-y}, \text{if} \ 0 < x ...
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Does $\lim\limits_{n \to +\infty} \left(\frac{n}{f(1)} \int_0^1 x^n f(x) dx \right)^n$ exists?

My question is having following one as a root. On one side, for $f : [0,1] \to \mathbb R$ continuous, one can prove that $$\lim\limits_{n \to +\infty} n \int_0^1 x^n f(x) dx =f(1)$$ On the other ...
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1answer
20 views

is the map $g:T\to [0,2\pi),\; e^{ir}\to r$ continuous?

$T=\{z\in\mathbb{C}: |z|=1\}$. Is the map $g:T\to [0,2\pi),\; e^{is}\to s$ continuous? Our teacher said, that $g$ is continuous on $T\setminus \{1\}$, what I don't understand. I tried to find a ...
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1answer
54 views

Prove there exists a unique local inverse.

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 ...
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0answers
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Range of values of $\frac {\epsilon}{\sup \delta}$ for the modified step function

Let the modified step function be defined on $[0,1]$ by : $f(x) = \begin{cases} \bigg( \dfrac {2^n+1}{2}\bigg )x - \dfrac {2^n-1}{2^n} ; & n \in \mathbb N~~ , \dfrac {2} {2^n+1} ...
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1answer
18 views

Problem with proving continuity of this function

As the title says I came across a problem with this function: $f:\mathbb{R}\to\mathbb{R}$ is continous $ f(x)=0 $, for $ x\in\mathbb{Q}$ So i want to prove that this works for every real number: ...
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1answer
23 views

Continuity of a function defined on the rationals

I'm presented with the following question, which I think is meant to be a precursor to material on completeness. Let $\alpha$ be an irrational number. Show that the function $f:\mathbb{Q} \to ...
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32 views

continuous (smooth) maps and group homomorphism

Consider a topological group $G$ (or smooth Lie group) and a topological space $M$ (or smooth manifold) and a group homomorphism $\phi:G\rightarrow Sym(M)$, where $Sym(M)$ is the symmetry group of M, ...
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1answer
39 views

If $x,y \in \bar{\mathbb{R}}$ then is $g(x,y)=xy$ continuous?

Suppose we assume the convention that $0 \cdot \infty =0$. If $\bar{\mathbb{R}}$ is the extended real line and $x,y \in \bar{\mathbb{R}}$, then is $g(x,y)=xy$ continuous? Why? I do not think it is as ...
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1answer
33 views

Being Careful with open sets

I have a function $f$ on [$0,1$]. I don't know whether on not it will be continuous at $1$ but I know if you pick any $0<\eta<1$ then it will be continuous on [$0,\eta$]. Surely this is enough ...
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1answer
11 views

Continuity of convex functions that have continuous restrictions to closed subspaces

Let $X$ be an infinite-dimensional normed vector space , let $U\subset X$ be an infinite-dimensional closed subspace, and let $f:X\to[0,\infty)$ be convex. Question: If the restriction $f|_U$ is ...
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3answers
44 views

Show that exist open ball B, such that $f(B)\cap g(B)=\emptyset$

$M,N$ metric space. Let $f,g:M\to N$ continuous in a point $a\in M$. If $f(a)\neq g(a)$, then exist a open ball B of center a, such that $f(B)\cap g(B)=\emptyset$. In particular, if $x\in B$, then ...