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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When does the quotient metric is equivalent to the quotien topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
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On a question about finite metric spaces

Let $(X,d)$ be a metric space such that every continuous function $f:X\to \mathbb R$ has a finite Image. prove that, $X$ is finite. I tried this: Let $x_0$ be arbitrary element of $X$ and define: ...
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a discontinuous function the square of which is continuous

give an example of a discontinuous function the square of which is continuous. The domain is $[0,1]$. I tried to use the indicator function of rationals, but its square is not continuous. EDIT:I am ...
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Need hints on how to start on this? [on hold]

Consider the function $f(x)=x^2$ Suppose we want to get error controls for $f(x)$ at $x_0 = 1$ with $ε = 0.1$. Show that $δ = 0.05$ is not sufficient, but $δ = 0.04$ is. Now try to do the same ...
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Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
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Show that this function is differentiable at all points [on hold]

n-sphotos-h-a.akamaihd.net/hphotos-ak-xta1/v/t34.0-12/11116109_10206718706905332_835173146_n.jpg?oh=baf1ad15e0f70703e5eb93818b61c9d1&oe=55401033&gda=1430237746_5bfaae4f8271730ad293b579ab0e93ab ...
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1answer
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Check whether the integrand is continuous when evaluating improper integrals

In order to evaluate improper integrals, I need to know whether the integrand is continuous between the limits of the integral. For the lower and upper limits, I believe you find out if it's ...
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2answers
43 views

Why the continuity of $f$ is not a necessary condition?

I am quite new to functions and continuity, and now I am reading the slides regarding the intermediate value theorem, which is related to continuity of functions. While reading, I found the ...
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1answer
25 views

What does continuity of a function mapping a topological space to a real line interval mean?

It makes sense for continuity to be defined on a function mapping a real line to a real line. Or how continuity is defined on a function between two topological spaces (every preimage of an open set ...
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1answer
16 views

Extending the definition of curve length

I know for continuously differentiable curves on closed interval $[a,b]$, the curve length is given by $\Lambda (\gamma)=\int_a^b |\gamma^{'}(t)|dt$. But what about curves such that $\gamma^{'}(t)$ is ...
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Construction of continuous function

Suppose a function $\phi:[0,1] \rightarrow [-1,1]$. Assume that the function $\phi$ has discontinuity at $x=1$ and $\phi(1)=0$. Question: Is it possible to construct a bijection and continuous ...
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On convergent sequences

Suppose that i have and open and surjective map between two metric spaces $\pi\colon X\to Y,$ and a sequence $(x_n)_{n\in \mathbb{N}}$ such that its image by $\pi$ converges. Is it true that ...
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1answer
38 views

Evaluating $\lim_{n\to\infty}\int_0^1x^nf(x)\,dx$. [duplicate]

Let $f$ be a continuous function on [0,1]. Evaluate $$\lim_{n\to \infty} \int_0^1 x^nf(x)dx$$ My approach : Consider $\int x^nf(x)dx = \frac{f(x)x^{n+1}}{n+1} - \frac{1}{n+1}\int x^{n+1}f(x)dx$ ...
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1answer
73 views

Spivak's calculus: Chapter 7 problem 18 d)

In cases (a) and (c) [where it was proven that such a number exists for a continous $f$ on $\textbf{R}$], let $g(x)$ be the minimum distance from $(x,0)$ to a point on the graph $f$. Prove that ...
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1answer
15 views

Monotone functions and distribution functions

I found this quote in a textbook on measure theory I'm studying: Let $f:[a,b] \to \mathbb{R}$ be an increasing function. Since $f$ has only countably many discontinuities, we may assume without ...
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2answers
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Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
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1answer
30 views

Characteristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
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“root” of a right-continuous function

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a right-continuous function such that $f(0) < 0$, $f(1) > 0$, and $f$ only changes sign once in the interval $[0,1]$. Suppose we define the "root" of ...
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1answer
27 views

Continuity and Directional Derivatives

Does every absolutely continuous function on a compact set possess a left and right hand derivative everywhere on its interior? Although the two need not be equal of course.
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Does the function $f(x) = \frac{x^2-1}{x-1}$ have any point discontinuity?

Since the domain of $f(x)$ is $(-\infty, 1) \cup (1, \infty)$ is there any point discontinuity in $f(x)= \frac{x^2-1}{x-1}$?
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Limit and Continuity dependency [on hold]

Whether someone knows any interesting example of dependency between limit and continuity of function? Thank you for any proposition
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What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
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1answer
41 views

Show that $f$ is continuous at exactly one point

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)= \begin{cases} 5x+7 & \text{ if } x \text{ is rational } \\ x+11 & \text{ if } x \text{ is irrational } \end{cases}$$ ...
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1answer
59 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
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2answers
40 views

Continuous function without a weak derivative

Let $f:\Omega\to\mathbb{R}$ be a continuous function. Is it necessarily true that $f$ has a derivative in the weak sense? That is, is there some $v:\Omega\to\mathbb{R}$ such that for every test ...
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continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
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2answers
33 views

Convergence of a sequence of integrals

I've tried expanding the hinted expression by using the definition from part (i) and choosing an X0 sufficiently large that |f(x)-l| < 1 but this doesn't appear to help very much at all. I've ...
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2answers
31 views

Continuity and measurability

this question concerns continuity and measureability. Am I right in thinking that if $f>0$ for all x then $\log(f(\lambda x)/f(x))$ is a continuous function for all $x$. Does this then mean that ...
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28 views

Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...
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Let $E$ be the space $L^1(\mathbb{R})\cap L^2 (\mathbb{R})$ equipped with the norm $\|u\|_E = \|u\|_1 + \|u\|_2$.

I am trying to solve this but I got stuck. Help needed. $E$ is a Bananch space. Let $f(x) = f_1(x) + f_2(x)$ with $f_1\in L^\infty(\mathbb{R})$ and $f_2 \in L^2(\mathbb{R})$. Check that the mapping ...
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Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
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1answer
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Showing that the functional $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$ is continuous

Suppose that we have the functional $L: L^2[a,b] \to \mathbb{R}$ , $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$. $f(x)$ is a well behaving, integrable function in $L^2[a,b]$. I want to show that this is a linear ...
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1answer
20 views

Can unbounded discontinuous functions be locally bounded?

Consider the function $$f(x) = \frac{x^3}{1+x^3}$$ Obviously this function is discontinuous at $x = -1$ therefore discontinuous on $\mathbb{R}$. Moreover, it is unbounded at the same point. Now, I ...
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1answer
37 views

Two functions equal in some point

I have two continuous functions $f,g$, $f(0) \lt g(0), f(1) \gt g(1)$. How do I prove without using "advanced" theorem (using only definitions of limit, continious functions and sup/inf definitions), ...
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Rationale behind a proof regarding a continuous function and an open ball

can I have the rationale for the first line of this proof? i.e. How did you know to start answering the question in this manner? I am guessing it is because you want to exploit the definition of ...
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1answer
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On the definition of uniform continuity over an interval.

I was reading some slides and I stumbled upon this definition of uniform continuity in an interval I am unsure on how to trace this back to the definition of uniform continuity that I know: A ...
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1answer
25 views

Proving isometry and continuity from a positive definite symmetric real matrix

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\epsilon$ be the Euclidean metric on $\Bbb R^n$ ...
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Continous frunctions problem

The problem says: f,g:[0;1]->[0,1] ,2 continous functions.They have the property that f(g(x))=g(f(x))). To solve: Both having the property of DARBOUX on the interval ,demonstrate that the numbers "c" ...
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1answer
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Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
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1answer
29 views

Proving the continuity of functions from one metric to another

I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The original ...
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37 views

Continuity of a peculiar function

I came across the question: Evaluate$f(x)=\lim_{m \to \infty}\lim_{n \to \infty}[\cos(n!\pi x)]^{2m}$. I simplified this to: $$f(x)= \begin{cases} 1 & \text{if $x \in Q$} \\ 0 & \text{if ...
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If function has a given limit then to prove that function is bounded.

How to Prove that if a function $f : A \to \Bbb R$ has a limit $l \in \Bbb R$ at $c \in L(A)$, then it is bounded in a neighborhood of $c$, i.e. there exists $M \in \Bbb R$ and $\delta > 0$ such ...
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1answer
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Let $f:[0,\frac{\pi}{2}]\to R$ be $f(x)=max\{x^2,cosx\}$.Prove $f(x)$ attains minimum at $x_0$ and is a sulution to $x^2=cosx$

I try to write $f(x)=\frac{1}{2}x^2+\frac{1}{2}cosx+\frac{1}{2}|x^2-cosx|$ and use the Extreme Value Theorem to show that $x_0$ exists in $[0,\frac{\pi}{2}]$, but I don't know how to show the seconde ...
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1answer
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Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
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1answer
70 views

$f(x) =\ln(2x^2 + 1)$ is continuous on $\mathbb{R}$

True or False The function $f : \Bbb R \to \Bbb R$ defined by $f(x) = \ln(2x^2 + 1)$ is continuous on $\Bbb R$. I know this condition that The function $f$ is continuous at some point $c$ of its ...
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Let $f,g$ be continuous from $\mathbb R$ to $\mathbb R$ [duplicate]

Let $f, g$ be continuous from $\mathbb R$ to $\mathbb R$, and suppose that $f(r) = g(r)$ for all rational numbers $r$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb R$?
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I need help finishing this proof using the Intermediate Value Theorem?

Let $f$ and $g$ be continuous functions on $[a,b]$ such that $f(a)\geq g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0)=g(x_0)$ for at least one $x_0$ in $[a,b]$. Here's what I have so far: Let $h$ be a ...
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Test for uniform continuity

Test for uniform continuity the function $ f(x, y) = (x^2 + y^2)^\alpha \sin{\frac{1}{x^2+y^2}} $ in $ \{ x^2+y^2 > 1\} $ If we consider $ \alpha < 1 $, then $ \lim_{\sqrt{x^2+y^2} \to ...
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2answers
45 views

Expanding a function

Is it possible to expand a function $$ f(x,y) = \dfrac{\sin (xy)}{\sqrt{x^2 + y^2}} $$ so it will be continuous on $\mathbb{R}^2$? Now, the denominator should not be equal to $0$, so for the domain, ...