Questions tagged [continuity]

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) or (general-topology)

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0answers
19 views

Interval of continuity over a jump

I thought I had this figured out but recently I was told some of what I think is conflicting information. I just want to clarify for my own peace of mind. Consider the piecewise function: $$f(x) = \...
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1answer
21 views

Real Analysis, countinuous function [closed]

If $f,g:[0,1] \rightarrow \mathbb{R}$ are countinuous functions that satisfy $\underset{x \in [0,1]}{\sup} f(x) = \underset{x \in [0,1]}{sup} g(x)$, then exists $x_{0} \in [0,1]$ such that $f(x_{0}) = ...
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1answer
13 views

Given the function,check the continuity at $(0,0)$ and find $f_x(0,0),f_y(0,0)$ if they exist

Given a function defined by : $$f(x,y) = \begin{cases} \frac{x^{3}+y^{3}}{x^{2}+y^{2}} & (x,y) \ne (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ Find $f_x(0,...
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1answer
54 views

Prove or disprove that $f\left(x\right)\leq 1,$ which is a continuous function so that $\int_0^x tf\left(2x-t\right){\rm d}t=\frac{\tan^{-1}x^2}{2}$

Prove or disprove that $f\left ( x \right )\leq 1,$ where $f$ is a continuous function so that $$\int_{0}^{x}tf\left ( 2x- t \right ){\rm d}t= \frac{\tan^{-1}x^{2}}{2}$$ Source: AoPS/@Ji123_ https://...
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1answer
31 views

Find the values of $a$ and $c$ to make $f(x)$ continuous at $x = 4$

Let $$f(x)=\begin{cases} 2cx, \quad x<4\\ x+a, \quad x=4\\ x^{2}-6, \quad x>4\end{cases}$$ I have to solve for $c$ and $a$. So far I got $5/4$ for $c$ and $4$ for $x$. Not sure how to go to $a$. ...
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1answer
23 views

Analysis - Prove a function that maps the unit Euclidean ball to R with bounded partial derivatives is uniformly continuous

I am stuck with this problem from my textbook and I cannot see the solution. I'm certain the solution is fairly simple and I am just missing the mark somehow. Any help would be appreciated. Suppose ...
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1answer
42 views

Discontinuous functions in real analysis. [closed]

Let $D=\{x_{1},x_{2},...,x_{n},...\}$. Can we find a function $f:\mathbb{R}\rightarrow\mathbb{R}$, which is discontinuous at each point of $D$?
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1answer
14 views

Semi-continuity for extended value function

Given a topological vector space $(X,\tau)$ a function $f:(X,\tau)\to\overline{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\}$ is said to be lower-semicontinuous at $x_0\in X$ if $$ f(x_0)\le\liminf_{x\to ...
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41 views

If $d$ is a metric, is it true that $d$ is lower semi-continuous?

A function $f:X \to \mathbb R$ is defined as Lower Semi-continuous if for every $x_n \to x$ $$ f(x) \leq \lim \inf_n f(x_n) $$ Let $(X,\tau)$ be a metric space with metric $\tau$. If $d$ is a metric ...
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2answers
21 views

Showing that the derivative function is not continuous w.r.t. the sup norm [duplicate]

Let $\mathcal{C}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f \text{ is continuous}\}$, $\mathcal{C}^{1}([-1, 1]) = \{f: [-1, 1] \to \mathbb{R}\mid f' \text{ is continuous}\}$, where both sets are ...
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1answer
66 views

About extended of continuous function [closed]

Assume we have an open interval $I\subset\Bbb R$ and a set $D\subset I.$ Let $f\colon D\to \Bbb R$ be a continuous such that $f$ can not be extended to a continuous on $I.$ What we can conclude from ...
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1answer
35 views

Real analysis - Problem on Continuity [closed]

Is it possible to have a function on a metric space which is discontinuous at every point of the metric space but the restriction of that function on a dense set is continuous?
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Uniform convergence over composition functions

Good evening; A friend who is studying real analysis asked me for help in this problem, I would like any clue of solution or complete solution, I also accept bibliographical references. Let $f: [0,1] \...
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1answer
53 views

Find a non-constant $f_n : \mathbb{R} \rightarrow \mathbb{R}$ so $f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}$ is continuous.

Find, for every $n \in \omega$, a non-constant function $f_n : \mathbb{R} \rightarrow \mathbb{R}$ so that $f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}$, defined as $f(x) = (f_n(x))_{n \in \omega}$, ...
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5answers
54 views

How to prove that $f(x, y)=\frac{2xy}{x^2 + y^2}$ is not continuous in $(0, 0)$ using an Epsilon-Delta proof?

In an assignment for the course Real-Analysis I need to proof that $f(x, y)=\frac{2xy}{x^2 + y^2}$ is not continuous in $(0, 0)$ using an Epsilon-Delta proof. However, me and my fellow students have ...
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1answer
29 views

Let $f_n$ sequence of continuous functions on $[a,b]$, $\lim_{n\to \infty}f_n(x)=f(x)$ uniformly on $(a,b)$. Show $f_n\mapsto f$ uniformly on $[a,b]$

Let $f_n$ sequence of continuous functions on $[a,b]$ and $\lim_{n\to \infty}f_n(x)=f(x)$ uniformly on $(a,b)$. Show that $f_n\mapsto f$ uniformly on $[a,b]$. I would like to know how to continue my ...
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1answer
29 views

Show that the following is only continuous at x=0

What is this question asking? How does rational and irrational affect it? What are the absolute values used for in this solution? Why is the series irrational when $c\ne0$ for any nonzero rational ...
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0answers
51 views

Showing that $\mathrm{Min}(f) = \min_{x \in [a, b]}f(x)$ is continuous

Let $\mathrm{Min}(f) = \min_{x \in [a, b]}f(x), f \in \mathcal{C}([a, b])$. I already know a proof for showing that a similar function, $\mathrm{Max}(f) = \max_{x \in [a, b]}f(x)$, is continuous. For ...
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1answer
85 views

Prove $f(x) = x^2 + 3$ is continuous at $x=3$

Prove that $f(x) = x^2 + 3$ is continuous at $x=3$. I have tried using $\delta = \sqrt{\epsilon + 9} - 3$. I tried to split $|x^2-9| = |(x-3)(x+3)|$ and tried to make $x+3$ in terms of $\delta$. But I ...
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1answer
47 views

Continuous Image of $S^{n-1}$ in $\mathbb{R}$

Let $f:S^{n-1}\to \mathbb{R}$ be a nonconstant continuous function. $S^{n-1}$ is Compact & Connected$\implies$ Its image is of the form $[a,b]$ in $\mathbb{R}$. For any $c \in (a,b)$: $\#\{f^{-1}...
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1answer
66 views

Fundamental Theorem of Calculus II at a point [closed]

FTOC part II states: Let $f$ be a continuous real-valued function defined on a closed interval $[a, b]$. Let $F$ be the function defined, for all $x$ in $[a, b]$, by $$F(x) = \int\limits^{x}_{a}{f(t)\,...
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1answer
66 views

Show that for every continuous function $f$, there exists a step function g such that $\int_a^b|f(x)-g(x)| dx< \epsilon$?

My idea is to take the integral and split it into the intervals of the step function but im not sure where to go from there and show that it is less than $\epsilon$. Thank you.
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0answers
12 views

find a value of a that makes the function continuous or demonstrate that no value of a is possible. [closed]

f(x)= {sin(ax) x<0x^2+4 x≥0 i am just lost just learnt this unit and a little confused with just one statement used to 2. Where to start is my main problem and how to set this up.
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37 views

How to prove that $f$ is continuous on $\mathbb{R^2}$ \ ${(0, 0)}$? [closed]

Let $f$: $\mathbb{R}^2$ $\longrightarrow$ $\mathbb{R}$ be defined by: $$f(x,y) =\begin{cases} {\frac{2xy}{x^2 + y^2}} &\text{for } (x, y)\neq (0, 0)\\ 0 &\text{for } (x, y) = (0, ...
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0answers
46 views

Why does the predicate reverse for the most powerful definition of continuity?

We can make three definitions of continuity, for a map $f$, which I know to be equivalent to each other, at least in the case of metric spaces: $f$ of any convergent sequence is convergent. Epsilon-...
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3answers
35 views

if $f(x,y)=\frac{xy^2}{x^2+y^6}$ and $f(0,0)=0$ show that $f$ is ubounded at any neighborhood of (0,0)

if $f(x,y)=\frac{xy^2}{x^2+y^6}$ and $f(0,0)=0$ show that $f$ is ubounded at any neighborhood of (0,0) but the restritions to straight line in $\mathbb{R}^2$ is cont. My attempt: **Proving continuity ...
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0answers
36 views

Is it true that any one-to-one continuous mapping between two topological spaces is a homeomorphism?

I was thinking if it always true that any one-to-one continuous mapping between two topological spaces is a homeomorphism? Are there any other conditions which imply homeomorphism?
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1answer
39 views

Let $F$ be a closed subset of a metric space $X$. Does there exist a continuous function $g : X \to R$ such that $F = g^{-1}({0})$?

Let $F$ be a closed subset of a metric space $X$. Does there exist a continuous function $g : X \to \mathbb{R}$ such that $F = g^{-1}({0})$? I am trying to produce a counterexample to this. Any ideas ...
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0answers
53 views

I have confirmed that the left integral diverges by the special value of the Lambert function in the equation $xe^{x^{2}}=1,$ anymore of it ??

Problem. Prove that $$\int_{0}^{\infty}\frac{{\rm d}x}{x^{2}e^{x^{2}}}= \infty$$ Indeed, that's true because $$\int_{0}^{\infty}xe^{x^{2}}{\rm d}x= 1/2\int_{0}^{\infty}{\left ( x^{2} \right )}'e^{x^{...
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1answer
31 views

A doubt about differentiability and increasing function

Suppose $f:(a,b)\rightarrow (a,b)$ is differentiable on $(a,b)$ and for an $x_{0}$ such that $a<x_{0}<b$ , $f'(x_{0}) >0 $ then is $f$ increasing in some neighborhood of $x_{0}$? I have seen ...
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1answer
34 views

Continuity of a function over a closed interval

The definition I have of continuity of a function $f:D \rightarrow \mathbb{R}$ at a point $c\in D$ is that $f$ is continuous at $c$ if for all sequeunces $(x_n)$ in $D$ s.t. $x_n \rightarrow c$, we ...
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2answers
41 views

Show that $g(x)=\frac{f(x)}{x} (x>0)$ is increasing

Let $f$ a continuous function for $x\ge 0$, $f'(x)$ exists for $x>0$, $f(0)=0$ and $f'$ increasing. Show that $g(x)=\frac{f(x)}{x} (x>0)$ is increasing. I would like to know if my proof holds, ...
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2answers
27 views

Proving a function is continuous through differentiability

I have the following function $f_n (x) = \sqrt{x^2 + \frac{1}{n^2}}$. I want to prove this is continuously differentiable. The method I have used seems like a bit of a shortcut but I was wondering if ...
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0answers
16 views

Monotonic and non monotonic function's integrability(Riemann) question

I was told that all functions that have a finite number of discontinuities, are Riemann-integrable. Then, I found a proposition that tells the following: All monotonic functions on closed interval $[a,...
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0answers
12 views

determine whether the limit of a function contains a moving discontinuity is in $L^1$.

I'm reading a paper A variational calculus for discontinuous solutions of systems of conservation laws from Alberto Bressan and Andrea Marson. I have a question at the very begining. Let $u^\epsilon(t,...
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0answers
48 views

Given two natural numbers $m, n$ so that $m> n.$ Prove that there exist any real numbers $x$ so that $2\sin nx\cos mx\geq 1$

Given two natural numbers $m, n$ so that $m> n.$ Prove that there exist any real numbers $x$ so that (unsolved in a case..) $$2\sin nx\cos mx\geq 1$$ Source: StackMath/@RiverLi_ Prove: $(\forall m, ...
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1answer
82 views

Is $C(\Bbb R)$ homeomorphic to $C([a,b])$?

Let $C(\Bbb R)$ be the space of all continuous self-maps of $\Bbb R$ in the compact convergence topology (which is same as compact-open topology) and $C([a,b])$ be the space of all continuous self-...
2
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1answer
31 views

Find a function discontinuous everywhere but $\ x = 0$

Find a function discontinuous everywhere but $\ x = 0$ After considering modifications to the Dirichlet function however I have not got anywhere. Is there a method to find such function or is it ...
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1answer
28 views

Continuity of derivative function [duplicate]

Let $f:X \to Y$ be a function such that it is differentiable on the interval $[x ,y] \subseteq X (x < y).$ If $a \in (x , y)$ and if $\lim\limits_{z \to a^+}f'(z)$ and $\lim\limits_{z \to a^-}f'(...
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0answers
18 views

Showing this multivariable function is uniformly continuous

Working on a problem from my class's textbook, and I'm stuck. The problem is seemingly easy, and I'm fairly certain I should be using the $\epsilon$-$\delta$ definition of uniform continuity, but I ...
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0answers
25 views

What is the dual space of the set of all riemann integrable functions

I was going through the dual of the basis of a vector space - which is essentially the set of linear functionals such that if $\{\alpha_1,\alpha_2,...,\alpha_n\}$ is the basis vectors then $\{f_1,..,...
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1answer
25 views

Maximum of the convex linear combination of continuous functions on a compact domain

I can't figure out a way to prove, or disprove, the following statement: let $(f_i(x))_{i\in I}, I=\{1,...,n\}$ be a collections of real valued, continuous functions defined on a compact domain, i.e. $...
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2answers
45 views

Confused about something that has to do with uniform continuity

I am confused about something that has to do with uniform continuity. The rectangles in the image above represent the regions formed by the following intervals, by the definition of uniform continuity ...
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0answers
8 views

Proving Continuity with polar coordinates (1st derivative)

Let's take a step back and consider the first derivative of $f(x,y) = x\,y\,\frac{x^2-y^2}{x^2+y^2}$. Here I take it that the first derivative $\partial_xf$ and $\partial_yf$ are continuous and hence ...
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0answers
10 views

Proving continuity with polar coordinates (2nd derivative)

This question is really bothering me: Given the function $f(x,y) = x\,y\,\frac{x^2-y^2}{x^2+y^2}$. Substituting polar coordinates: $f(r,\varphi) = \cos\left(\varphi \right)\,\sin\left(\varphi \right)\,...
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1answer
48 views

$f:\mathbb R\to \mathbb R$ s.t. $f(x) = 1$ for $x\in\Bbb Q$ and $f(x) = 0$ for $x\notin \Bbb Q$ is discontinuous

$f:\mathbb R\to \mathbb R$ s.t. $f(x) = 1$ for $x\in\Bbb Q$ and $f(x) = 0$ for $x\notin \Bbb Q$ is discontinuous. So the function above is obviously discontinuous, and I am looking to prove the same. ...
2
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0answers
28 views

An less commonly seen sufficient condition for convexity

I was reading a proof in which the author claimed that, for a function $f(x)$ on $(0,1)$ which is continuous and strictly increasing, the following condition implies that convexity holds. Let $A > ...
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0answers
28 views

Estimates of the remainder in Taylor's Theorem (Wikipedia)

Read the first two lines of this section I think it is sufficient to say that $f$ is $(k+1)$-times differentiable in an interval $I$ containing $a$ with $$q \leq |f^{(k+1)}(x)| \leq Q$$ for all $x \in ...
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0answers
20 views

Let h(x)=g(f(x)). Is there a limit at limx→4h(x) or explain why the limit does not exist? [duplicate]

Context: This was for a school Calculus 12 assignment and my teacher didn't explain it very well, especially since I looked on khan academy and it was explained very differently. I also want to learn ...
0
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1answer
21 views

continuity with polar coordinates same as sequential continuity?

I got a function: $f(x,y)= \begin{cases}\frac{x^3-3\,x\,y^2}{x^2+y^2} &(x,y) \neq 0\\ 0&(x,y) = 0 \end{cases}$ substitution polar coordinates ($x = r\,\cos(\varphi)$, $y = r\,\sin(\varphi)$) I ...

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