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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The set of continuity of a pointwise limit of continuous functions

Let $\{x_n(t)\}_{n=1}^{\infty}$ be real a sequence of continuous function from $[0,1]$ to $\mathbb{R}$, and $\{x_n(t)\}_{n=1}^{\infty}$ converges pointwise to $x(t)$ i.e. $\lim_{n \to \infty} x_n(t) ...
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Is it true that a mapping between metric spaces is continuous iff the image of every open set is open?

Just want to change Rudin theorem 4.8 a bit and see if this works. The original theorem is ... $f$ is continuous iff $f^{-1}(V) $ is open in $X$ for every open set $V$ in $Y$. If I change the ...
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3answers
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Continuity of function consisting of an infinite series.

Let $f(x) , 0\leq x\leq 1$ be defined by, $$f(x)=\sum_{n=1}^{\infty}\frac{1}{(x+n)^2}$$. Show that $f$ is continuous on $[0,1]$ and that, $$\int_0^1f(x)dx=1$$. I have never dealt ...
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1answer
406 views

Is the Sinc function continuous?

Is $\frac{\sin x}{x}$ a continuous function or is it not? I am confused with the fact that at zero it cannot be defined yet the limit surely exists. So, the question of its continuity arises.
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Continuity in $\mathbb R$ results in continuity in $\mathbb R^2$; Proof?

During studying of proof of some other theorem, I faced with the claim (without proof): since $f(x,t)$ and $g(x,t)$ are continuous functions [$f,g:\mathbb R^2 \rightarrow \mathbb R$] thus the ...
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1answer
12 views

Upper semi-continuity results

I have recently been introduced to the notion of upper semi-continuity on a metric space $X$. Please advise on the following queries: If $f:X \rightarrow \mathbb{R}$ is upper semi-continuous and ...
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4answers
724 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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2answers
65 views

Is my proof for this limit correct?

I want to prove that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to 2. Let $a_0$ = $\sqrt{2}$ $a_n$= $\sqrt{2+a_{n-1}}$. Then, proving that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to ...
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1answer
11 views

Limiting and continuous about one function

I have a function which is \begin{equation} F(x)= \begin{cases} f(x) & x \in [\underline{x},\bar{x})\\ \\ f(\bar{x}) & x=\bar{x} \end{cases} \end{equation} The function $f(x)$ is strictly ...
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1answer
31 views

which hypothesis for boundedness of this function

Let $v:[0,\infty)\rightarrow \mathbb{R}_+$ be a positive function such that $$\exists T,q>0\,\,s.t.\,\, \forall t\in[0,\infty),\,\,\int_t^{t+T} v(\tau) d\tau \le q$$ I'm looking for the "less ...
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196 views

Continuity of the sum of continuous functions

Let $X$ be a topological space and $f:X\to \mathbb{R}$ and $g:X\to \mathbb{R}$ be continuous functions. How do I show that $h:X\to \mathbb{R}$ where $h:=f+g$ is continuous, would prefer to use the ...
4
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2answers
369 views

Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.

So we want to find a $\delta>0$ such that for all $2-\delta<x<2+\delta$ , we will have $4-\epsilon<x^{2}<4+\epsilon$ for all $\epsilon>0$ . If we can find a way to express ...
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1answer
48 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
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1answer
60 views

Alternative Proof of the Extreme Value Theorem

I have proven the Boundedness Theorem for continuous functions and would now like to prove the Extreme Value Theorem; that is, show that the upper bound is indeed attained for continuous functions. I ...
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227 views

Solve this functional equation:

Functional equations such as this one appear only once every several years on exams, so I feel it's hard to have a sure-fire way to approach the problem, unlike, say, solving a series convergence ...
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1answer
49 views

How can I prove that this function is continuous in (0,0)? [on hold]

I have this function: $$ \lim_{(x,y)\to (0,0)} = \frac{2(1-\cos(xy))+\arctan(x^4)-x^2(x^2+y^2)}{(x^2+y^2)^\alpha} $$ I have to find which $ \alpha$ makes the function continuous. But my first problem ...
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1answer
33 views

Positive derivative on [0,1] implies a continuous derivative on [0,1]

If a real-valued function F defined on [0,1] is differentiable with positive derivative f everywhere on [0,1], can we conclude that f is continuous?
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1answer
27 views

Distance of a point to a subset.

Let $(M,d)$ be a metric space. For a subset $A\subseteq M$ we define the distance of a point $x$ to $A$ as $$\alpha_A(x):=\operatorname{dist}(x,A):=\inf_{y\in A}d(x,y)$$ Prove that: ...
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1answer
28 views

Continuity of composite functions

The continuity theorem for composite functions states that if $f(x)$ is continuous at $x = a$ and $g(x)$ is continuous at $x = a$ , then the composite function $f\circ g$ and $g\circ f$ are also ...
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Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
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2answers
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Is homeomorphic image of closed bounded subsets of metric spaces , also closed bounded in the homeomorphic image metric space?

Let $X$ , $Y$ be homeomorphic metric spaces with homeomorphism $f$ , then is it true that for any closed bounded subset $A$ of $X$ , $f(A)$ is also closed and bounded in $Y$ ?
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If $f(A)\to A^{-1}$, prove that $f$ is continuous.

Let $f \colon GL_{n}(\mathbb{R})\to GL_{n}(\mathbb{R})$ be a function which maps $A\mapsto A^{-1}$. Prove that $f$ is continuous. $GL_{n}(\mathbb{R})=\det^{-1}(\mathbb{R}\setminus\{0\})$ is ...
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If for every $a > 0$, $u \in C^\infty([a,\infty))$, then is $u \in C^\infty((0,\infty))$?

Suppose that for every $a > 0$, $u \in C^\infty([a,\infty))$. Does this imply that $u \in C^\infty((0,\infty))$? I think it is true when we just work in $C^0$, but with $C^\infty$ you need to ...
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1answer
111 views

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$.

Prove $f(x,y,z)=e^{iy+z}$ is continuous on $\mathbb R^3$. I have already proved that other functions are continuous by using that $f, g$ are continuous implies $f+g$ and $fg$ are continuous. ...
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1answer
480 views

Prove that if f is a continuous strictly monotone function defined on an interval, then its inverse is also a continuous function.

There is a theorem on continuous function that goes as follow: If f is a continuous strictly monotone function defined on an interval, then its inverse is also a continuous function. I have quite an ...
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0answers
26 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
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33 views

homeomorphism as a result of other homeomorphisms

If $$B = \bigcup_{R>0} B_R$$ and all the identities $$\operatorname{id}_R : (B_R,d_1) \rightarrow (B_R,d_2)$$ for $R>0$ are homeomorphisms, then is $$ \operatorname{id} : (B,d_1) \rightarrow ...
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1answer
34 views

finding and proving where function is…

So I have this function: $ f(x) = \begin{cases} ( 2 \sqrt{-1-x}-1)^{\frac{1}{4^{-x}-16}} & \quad \text{if } x<{-2}\\ - \frac{\pi}{4}x & \quad \text{if } -2\leq x \leq 1 \\ \frac{\sin{(\pi ...
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1answer
54 views

Prove that the difference of continuous and monotonically increasing functions has continuous variation

Let $G:[0,\infty)\to\mathbb{R}$ be continuous and $$V^1_t(G):=\sup\bigcup_{n\in\mathbb{N}}\left\{\sum_{i=0}^{n-1}\left|G_{t_{i+1}}-G_{t_i}\right|:0=t_0\le\cdots\le t_n=t\right\}$$ be the variation ...
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1answer
38 views

Intuition on the Topological definition of continuity, considering the special case of the step function.

I'm trying to get an intuition for open sets and topological reasoning in general. One example I want to understand is the step function, and specifically why it would be considered discontinuous ...
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173 views

To find continuous functions on $\mathbb R$ which preserve certain algebraic structures

Can we determine all non-constant continuous functions $f:\mathbb R \to \mathbb R$ such that for every subgroup $G$ of $(\mathbb R,+)$, $f(G)$ is also a subgroup of $(\mathbb R,+) $ ? And ...
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30 views

Relation between $\lim_{a \to 0}\int_a^T u(t)$ and the Lebesgue integral $\int_0^T u(t)$

Let $u\colon (0,T] \to \mathbb{R}$ be function with $u \geq 0$ everywhere and $u$ is continuous on $[a,T]$ for every $a > 0$. Suppose that the limit $$\lim_{a \to 0}\int_a^T u(t) \;dt ...
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1answer
38 views

When can I take $\lim_{a \to 0}\int_a^T u$?

Suppose I have a function $u:(0,T) \to \mathbb{R}$ which is integrable over $[a,T]$ for every $a > 0$, and I have the results $$\int_a^T u = U(T)-U(a)$$ for such $a$. When am I allowed to conclude ...
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1answer
26 views

Help with vacuous continuous function (please)

i have a question that's been bugging me for the past two days. The definition of a function that is continuous at some point $a$ of it's domain, states: $f$ is continuous at $a$ if $$\lim_{x\to a} ...
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27 views

Strictly monotonic increasing function with a closed domain and range

Let $a,b,c,d \in \mathbb{R}$ with $a<b$, $I = [a,b]$. Let $f: I \rightarrow \mathbb{R}$ be a monotonic, strictly increasing function. Also $c<d$ and $f([a,b]) =[c,d]$ a) Proof that $f$ is ...
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1answer
31 views

The image of the inverse of a continuous function

First of all I'm not sure if my title is correct with the question, I find it hard to really get about what kind of set this question is about. It would be very helpful if someone could explain this ...
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30 views

Continuity on a given set

Given a set A subset of R and B subset of A, can we define a function which is continuous on B but discontinuous in A-B(where A-B does not contain any isolated point) Given a set A subset of R and B ...
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Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
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1answer
28 views

Upper semi-continuity proof for topological spaces

Hi does anyone have any idea or a possible hint for a proof of the following result: Consider asymmetric norm $p$ on $\mathbb{R}$ given by $p(t) = t^{+}$, for $t \in \mathbb{R}$. Show that if $(X, ...
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59 views

prove continuity

Let $ f:\Bbb R \to \Bbb R $ satisfy the property $ f(x+y)=f(x)+f(y)$ for all $x,y$ in $ \Bbb R $ I have to show that 1)$f(0)=0 , f(-x)=-f(x),$ for all $x$ in $\Bbb R$, and $f(x-y)=f(x)-f(y)$ $y$ in ...
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Spectral Measures: Norm

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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72 views

Show that $f$ is continuous mathematically.

Let $f:[0,\infty)\to \mathbb{R}$ be given by $f(x)=\sqrt{x}$. Show that it is continuous. This is taken from Example 3.7 on <link> page 22 on the paper. It has shown that it is continuous at ...
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92 views

Looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points

I am looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points ; please help , thanks in advance .
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Is a function $f \in \mathbb{C}^{ \infty}[0,l]$ always in $L^2(0,l)$?

I was trying to find a function that is not in $L^2(0,l)$ but that it is in $\mathbb{C}^{\infty}[0,l]$ for l>0. But if the function is continuous at both sides of the interval then it is integrable, ...
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Is there a function Lipschitz on the right of every point, but everywhere discontinuous?

Today I came across the following definition: Definition: A function $f:[a,b] \to \Bbb C$ is Lipschitz to the right of $t_0 \in [a,b]$ if exists $L>0$ such that $|f(s+t_0)-f(t_0^+)| <Ls$ for ...
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2answers
56 views

Is this Function differentiable and continuous at x=0? [closed]

Is $f(x)$ continuous and differentiable at $x = 0$ ? $$f(x) = x(\sqrt{x} - \sqrt{x+1})$$
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If you have a function $f(x)=\frac{x^2}{x}$, then is the function continuous at x=0?

If you have a function $f(x)=\dfrac{x^2}{x}$, then is the function continuous at $x=0$? On one hand, if you simplify it and end up with $f(x)=x$, it is continuous at $0$, but if you keep it in its ...
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1answer
52 views

Anywhere I integrate $f_n$, the integral approaches $f$. Is $\lim_n f_n = f$ a.e.?

Something tells me this is obvious... I have a bunch of functions: $f,f_n:\mathbb{R}^2\rightarrow \mathbb{R}$, all integrable. Also, $f$ is continuous. I also have a family of sets, $\mathcal{G}$ ...
2
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1answer
46 views

Every continuous map of a closed interval into itself has a fixed point

The Question: Please show this theorem: Let $f: I=[a,b] \rightarrow \mathbb{R}$ be a continuous map such that $f(I) \supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a ...
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293 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...