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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limits changed F(g(x))=F(lim G(x))

lim x tends to 0(F(G(x))=F(lim x tends to 0(G(x))) I have seen this step in a derivation of a result which is not the point of interest here. The book wrote the reason for it was that it is when F ...
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Upper semicontinuous function and equivalent statements

Problem Let $f:\mathbb R^n \to \overline{\mathbb R}$, then the following statements are equivalent: (1) $f$ is upper semicontinuous; (2) for every $t \in \overline{\mathbb R}$, $\{x \in \mathbb ...
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Prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$

I need to prove that $\mathbb{R} \times S^1$ is homeomorphic to $\mathbb{R^2} \setminus \{(0,0)\}$. I define the map $h:\mathbb{R} \times S^1 \to \mathbb{R^2} \setminus \{(0,0)\}$ by ...
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Relation between continuous maps and convergence of sequences

I am studying metric spaces and I know that in a normed space $E$ a map $T:E \to E$ is contínuous if and only if $T(x_n) \to T(x)$ for every convergent sequence $x_n \to x$ in $E$. In my notes there ...
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1answer
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If a function is bounded and the variable is bounded, is the function continuous?

Suppose you have a function $f:C\to \mathbb R$ where $C$ is closed and bounded interval and $f$ is bounded. Does that mean $f$ is continuous? I know the other way around (if $f$ is continuous, $f$ ...
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3answers
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Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
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1answer
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Proving Continuity and equivalence

I have posted ths on the Quant Finance page as it is part of a QF problem but realised I may get a swifter response here! Iam working on a problem where I have successfully reduced a version of Black ...
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1answer
63 views

If a function is both upper and lower semicontinuous, does it have to be continuous?

I am looking for an example of a function which is both upper and lower semi continuous but is not continuous. I have an example: $$f(x):=\begin{cases} 1 & \mathrm{if}\; x < 1,\\[7pt] ...
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1answer
38 views

Question on continuity with [x]

given the function $f(x)=\frac{2[x]}{3x-[x]}$ the question is to find continuity of the function at $x=1$ and $x=\frac{-1}{2}$ note: [x] denotes the largest integer which is less than or equal to x. ...
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Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ ...
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Measure derivatives and the chain rule

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$ such that $\mu << \lambda$. Prove that $\displaystyle \int D(\mu,\lambda,x)^2 d\lambda x= \int D(\mu,\lambda,x)d\mu x$. Is it ...
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1answer
29 views

how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on ...
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52 views

How many continuous functions are differentiable? [duplicate]

Consider the set of continuous functions $\mathbb{R} \to \mathbb{R}$. I assume that the subset that are not everywhere differentiable accounts for almost all of them. Is this true? What is the precise ...
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Question about writing a proof with continuous functions [duplicate]

How would I write a proof for this example? We know that all polynomial functions on the reals are continuous by using the sequential definition of continuity. In particular, we know that the ...
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1answer
35 views

Does differentiability imply absolute continuity? [duplicate]

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a function which is (i) differentiable at all $x \in (a,b)$ (ii) the right-derivative at $x=a$ exists and the left-derivative at $x=b$ exists. Does it ...
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Direct proof of uniform continuity on compact set

I've looked in several books for a direct proof of the theorem that says if a function is continuous on a compact set, then it is uniformly continuous. I've only found proofs that argue by ...
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1answer
520 views

Where is the error in my proof that all derivatives are continuous?

I know that this can not be true due to counter-examples but I don't know where the error in my reasoning is. Assumption: If $f(x)$ is differentiable in $\mathbb{R}$ then the derivative $f'(x)$ is ...
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2answers
98 views

Epsilon-Delta continuity definition for straight lines parallel to axes

I am taking a course on real analysis online and I encountered the $\epsilon-\delta$ definition for a function to be continuous. But I wonder if I can apply it to functions which are straight lines ...
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1answer
34 views

A continuous function that attains neither its minimum nor its maximum at any open interval is monotone

Let $f: \mathbb R\to \mathbb R$ be a continuous function such that $f$ attains neither its minimum nor its maximum at any open interval $I \subseteq \mathbb R$ , then how to prove that $f$ is ...
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1answer
33 views

Are the family of given nice functions $f\subset C^0(I,[0,1])$ equicontinuous?

The family of continuous functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows: \begin{equation} f(y) = \begin{cases} 0, &l(y)<\rho \\ ...
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1answer
44 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
33 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
34 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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111 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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problem on continuity [closed]

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
21 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
41 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 ...
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0answers
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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
42 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 ...
2
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2answers
48 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
56 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
136 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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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
72 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
30 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 ...