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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Characterization of Sobolev Space

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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1answer
25 views

Are the family of functions $C^0(I,[0,1])$ equicontinuous?

I searched but couldn't find. Are the family of continuous functions $C^0(I,[0,1])$ equicontinuous for the finite interval $I\subset\mathbb{R}$? To claim this, I guess for every $\epsilon>0$ ...
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1answer
31 views

What is the difference between the terms smooth, analytical e continuous?

I saw the following (“roughly speaking”, like the author says) definition of a Lie group in ‘Group theory in Physics’, by Wu-Ki Tung: “Roughly speaking, a Lie group is an infinite group whose ...
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1answer
49 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
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1answer
46 views

Is $f(x) = \left(x^2 + \lfloor x^2\rfloor\right) \sin (2 \pi x)$ continuous?

Let $f \colon [0, \infty) \rightarrow \mathbb{R}$ is given as $f(x) = \left(x^2 + \lfloor x^2\rfloor\right) \sin (2 \pi x)$. Then can we comment on the continuity of $f$? Here $\lfloor x\rfloor$ is ...
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1answer
35 views

(Dis)continuity of function in $R^2$

$$f(x,y) = \begin{cases} a+2x^{2}-b(y-c), & x^{2}>2+x\wedge y<6\\ 3+cx-y, & else \end{cases}$$ $f(x,y)$ is continuous on $R^2$ if $a=-3, b=1, c=2$ I think it's true: insert ...
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1answer
33 views

What is the definition of this set of absolutely continuous function

I know that $$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ $$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d ...
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1answer
25 views

Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
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4answers
107 views

If $f(x)$ is discontinuous at $x=0$, can $\int_{-1}^1 f(x)dx$ exist.

I am interested in the reasoning. All help is appreciated
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3answers
59 views

problem on continuity [on hold]

For $x>0$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f:[0,\infty)\rightarrow\mathbb{R}$ be given by $f(x)=[x^2+[x^2]]\sin(2\pi x)$. Then $f$ is continuous at $2$ or ...
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1answer
13 views

In the semi linear uniform space

In the semi linear uniform space, If $f$ is a function from $(X ,Γ_X)$ to ($Y,Γ_Y)$ where $f(x_n)$ converges to $f(x)$ whenever $x_n$ converges to $x$,show that $f$ is continuous at $x$.
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1answer
26 views

Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...
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2answers
39 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
3
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0answers
29 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
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2answers
49 views

Is my proof correct? Finite-dimensional normed vector spaces

I'm trying to prove that every finite-dimensional normed space is topological isomorphic to $\mathbb{R}^n$. Let $(E,\|\cdot\|_E)$ such that $dimE=n$ and let $$ T:\mathbb{R}^n\to E\\ x\mapsto ...
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1answer
27 views

Continuity of a map to a Frechet space

Let $(A,\| \cdot \|)$ be a normed space and $B$ be a Frechet space equipped with a family $\{ p_k \}_{k \in \mathbb{N}}$ of seminorms. Let $\phi: A \to B$ be a linear transformation satisfying the ...
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1answer
39 views

Requirements for integration by parts/ Divergence theorem

In order to use the integration by parts formula(or more generally the divergence theorem) for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v ...
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2answers
47 views

Requirement for continuity of unit normal vector

When considering a subset $\Omega \subset \mathbb{R}^{n}$. If we consider $\nu$, the outward unit surface normal to $\partial \Omega$, what are the requirements of $\partial \Omega$ which will ...
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1answer
21 views

Sequence problem dealing with continuity and convergence.

I need help in this question. I figured out a way to solve the question but not sure the proof is valid. This is the question, Given $a \in\mathbb{R}$, and a function ...
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1answer
41 views

Continuity basic understanding

I have been asked to figure out if they are continuos or discontinues or left or right con/discon for the point -2. -1. 0. 1. 2. , where the function g(x) has domian[-2,2]. I just do not get it. As ...
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1answer
38 views

Intuition behind homeomorphism from $B((0, 0), 1) \to \mathbb{R^2}$

In my notes I have that the following function is a homemorphism from $B((0, 0), 1) \to \mathbb{R^2}$ $$h(x, y) \to \frac{f(\sqrt{x^2 + y^2})}{\sqrt{x^2 + y^2}} (x, y)$$ where $f = ...
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3answers
55 views

True or False Question About Functions [closed]

If $f(1)>0$ and $f(3)<0$, then there exists a number $c$ between $1$ and $3$ such that $f(c)=0$. I'm not sure how to solve this question. Thanks in advanced!
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4answers
135 views

Real Analysis: Showing $f: \Bbb Q \to \Bbb Q$ is continuous

The following is all working in $\mathbb{Q}$, not $\mathbb{R}$. I am working with the function $f: \mathbb{Q} \to \mathbb{Q}$ defined piece-wise by $f(x)=-1$ if $x^2<2$ $f(x)=1$ if otherwise I ...
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2answers
45 views

Continuity of a multivariable function with “parts”

I'm trying to solve if $f$ is continuous: $$ f(x,y) = \begin{cases} x^3 + y^3 &\text{if }y>0 \\ x^2 &\text{if }y ≤ 0 \end{cases} $$ I have seen that $$\lim_{(x,y) \to (0,0)} ...
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0answers
17 views

Intermediate Value theorem application 3

How do we use the intermediate value theorem repeatedly to find the root of the equation $x^{6}$- $x^{5}$ + $2x^{4}$ - $2x^{3}$ - $3x^{2}$ - $2x$ -$1$ = $0$, which lies on the interval [$1$,$2$] up to ...
2
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1answer
30 views

Continous surjective map from $S^1$ to $S^n$

Is there any continous surjective map from $S^1$ or $[0,1]$ onto $S^n$, for some $n\geq 2$. Thank you.
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5answers
71 views

Example of continuous increasing and decreasing functions that don't intersect.

I am looking to describe two continuous functions. One of them is strictly increasing on the real line and one of them is strictly decreasing on the real line. I want to describe these functions in ...
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1answer
35 views

Discontinuous Differential Equation

Is there a solution to the following equation? If so, what is it? $$\frac{df}{dt}= \begin{cases} -t, & f\geq 0\\ t, & f<0 \end{cases}$$ Thanks.
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1answer
29 views

Proving continuity of Thomae's function at irrational points

Let $h:\mathbb R^+ \to \mathbb R$ be a function such that $h(x)=0$ for every irrational $x$ and for any rational number in $\mathbb R^+$ of the form $\dfrac mn ,$with g.c.d.$(m,n)=1$ , we define ...
3
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1answer
45 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
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2answers
31 views

Determine intervals on which s(t) =equation

Determine the intervals on which $$s(t) = \frac{|t^2-2t - 3|}{t + 1}$$ is continuous. Hint: Use continuity checklist and check left and right continuity of proposed intervals which include ...
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1answer
27 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
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4answers
593 views

How does this discontinuity occur in evaluating a nested square root?

This question is based on a comment I made on a question likely to be closed. Let $$y=\sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \sqrt {x+ \dots}}}}}$$ be the classic nested square root which has ...
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1answer
20 views

Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
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2answers
28 views

Prove that $\exists \delta >0$ s.t.$ f(x)>0$, $\forall x \in (a-\delta,a+ \delta)$

Given $a\in \mathbb R$ and a function $f: \mathbb R \to \mathbb R$, prove that if $f$ is continuous at $a$ and $f(a)>0$, then $\exists \delta >0 $ s.t. $f(x)>0$, $\forall x \in (a-\delta,a+ ...
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0answers
37 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
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0answers
23 views

Binary search (bisection method) - is it worth checking continuity

I am implementing a rather simple matlab code, that gets a function $f$ and 3 real numbers $a,b, \epsilon$ where $\epsilon >0$ is a very small positive number (for instance, no larger than ...
0
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1answer
32 views

Continuity of a complex function $f$ at $a$ implies $\lim \limits_{z\to a}(z-a)f(z)=0$

Assume the complex valued function $f$ of a complex variable is continuous at $a\in\mathbb C$. How can we see that $$\lim_{z\to a}(z-a)f(z)=0$$
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2answers
43 views

showing the function is continuous at a point using $\epsilon$ and $\delta$

I have this question: Use the definition of continuous function with $\epsilon$ and $\delta$ to show that the function $f$, defined as $$f(x)=\begin{cases}0&\textrm{if } x=0 \\x \sin\frac{1}x ...
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2answers
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
36 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
139 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
62 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 ...
3
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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. ...