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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What kinds of functions have fixed points?

Among continuous functions, can we characterize those which have fixed points and those which do not? Geometrically, these are the functions that intersect the line $f(x) = x$. Is that the most ...
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110 views

Computing the volume of this weird object,

Let $f: [-1,1] \to \mathbb{R}$ be a continuously differentiable function such that $f(-1) = f(1) = 0$ and $0<f(x)\le 1$ for all $x \in (-1,1)$. Let $S$ be the surface in $\mathbb{R}^3$ obtained by ...
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Condition for function $f(x)=\frac{(1-x)^{-1/2}-(1+x)^{1/2}}{(1-\frac{x}{2})^{-1/2}-(1+\frac{x}{2})^{1/2}},(x\neq0)$ to be continuous at $x=0$.

This function is not continuous at $x=0$. I know that function (in the example) is continuous if $$\lim\limits_{x\to0^-}f(x)=\lim\limits_{x\to0^+}f(x)=f(0)$$ and limits and $f(x_0)$ must be defined. I ...
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39 views

Coninutity of this function in interval $(0,1)$ [closed]

Let $f(x)$ be the function defined on the interval $(0,1)$ by $$ f(x) = \begin{cases} x(1-x) \quad\text{if}\quad x \in \Bbb Q \\ \frac{1}{4}-x(1-x) \quad\text{if}\quad x \in ...
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34 views

Continuity of increasing function [duplicate]

If $f$ is an increasing function over the reals, given a number $M$, is it always possible to find some $x \ge M$ such that $f$ is continuous at $x$? This seems like it should be intuitively true but ...
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63 views

Show that the function $(x^2+y^4)f(x,y)+f(x,y)^3=1$ is $C ^ 1$ class.

Consider $f:U\subset\mathbb{R}^2 \rightarrow \mathbb{R}$ a continuous function in open set $U$. Show that $(x^2+y^4)f(x,y)+f(x,y)^3=1$ is $C ^ 1$ class, for all $(x,y) \in U$. I think we can use the ...
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42 views

What are the restrictions on using substitution in integration?

* One photo is equal 1000 words. * Integration done by substitution $u=\tan x$. Integration done by substitution $u=\tan {x\over 2}$. The source function is a continuous positive function ...
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3answers
69 views

Is there any function $f:\mathbb R \rightarrow \mathbb R$ such that it is only continuous at rational numbers?

Is there any function $$f:\mathbb R \rightarrow \mathbb R$$ such that it is only continuous at rational numbers?
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1answer
86 views

$f:\mathbb R\to \mathbb R$ continuous, $f(f(0))=0$ so there exists $a \in \mathbb R$ such that $f(2a)=3a$

Let $f:\mathbb R\to \mathbb R$ continuous such that $f(f(0))=0$. Prove that there exists $a \in \mathbb R$ such that $f(2a)=3a$. Well, I figured that in such exercises, I should define a new function ...
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29 views

If $f$ is continuous and piecewise $C^1$ and $f'$ is bounded a.e., is $f$ Lipschitz?

If $f$ is continuous and piecewise $C^1$ on $\mathbb{R}$ (only a finite number of pieces) and $f'$ is bounded a.e., is $f$ globally Lipschitz? So $f$ is only not differentiable in a finite number of ...
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46 views

continuous and monotonic function [duplicate]

If there is a function that continuous in a interval monotonic in the same interval Does it mean the function is also differentiation function in the interval?
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30 views

continuity of inverse function

I studied derivative of function ${f^{-1}}'(y)=\frac1{{f}'(x)}$ When I tried above proof , it needs continuity of inverse function At this point , I have a question $f$ is continuous on D , then ...
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28 views

Is this multivariable function continuous?

My function is: $$f\left(x,\:y\right)\:=\:y\left(sin\left(\frac{1}{x-1}\right)\right)\::\:x\:\ne 1$$ $$f\left(x,\:y\right)\:=\:0\::\:x\:=1$$ The question sounds like: "Are this functions continuous?" ...
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Is the continuous extension theorem true when the range space of $f$ is not complete?

So the problem is Exercise $13$, Chap. $4$ of Principles of Mathematical Analysis by Rudin: Problem Let $E$ be a dense subset of metric space $X$, and let $f$ be a uniformly continuous real function ...
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31 views

Continuous of function in a point

Given the function $$f(x,y)=\frac{1-\cos(2xy)}{x^2y^2}$$ I want the function to be continuous in $(0,0)$. If I assume that the limit when $x\rightarrow0$ equals to the limit when $y\rightarrow0$, I ...
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103 views

Continuous bijective function between the same topology that is not a homeomorphism.

I know there are many examples when the domain and co-domain do not coincide. Taking the identity on $X$ from $(X,\tau_1)$ to $(X,\tau_2)$ when $\tau_2$ is coarser than $\tau_1$ gives an infinite ...
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1answer
40 views

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ ...
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61 views

Prove that $U-f(U)$ is an open set.

Let $(X,d)$ be a compact metric space. Let $f:X\to X$ be continuous. Fix a point $x_0\in X$, and assume that $d(f(x),x_0)\geq 1$ whenever $x\in X$ is such that $d(x,x_0)=1$. Prove that $U\setminus ...
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44 views

show that the function $\{x_n\}\mapsto \sum_{n=1}^\infty 2^{-n}x_n$ is continuous

This problem comes from an old Preliminary exam: Consider the space $[0,1]\times [0,1]\times \cdots$ (the countably infinite product of $[0,1]$ with the product topology) An element of $X$ may be ...
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38 views

Trying to prove that if $f:[a, b]\to[s, t]$ is monotone then $f$ is continuous

I'm trying to prove that if $f:[a, b]\to[s, t]$ is monotone (and its image is closed interval) then $f$ is continuous. My attempt: I say wlog, $f$ is increasing. I know that a monotone function only ...
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1answer
113 views

IMC 2008 first problem first day. Finding continuous functions so $x-y\in \mathbb Q \implies f(x)-f(y)\in \mathbb Q$

I would like an alternate solution and proof verification for the following problem: Find all continuous functions $f:\mathbb R \rightarrow \mathbb R$ so that if $x-y$ is rational then $f(x)-f(y)$ is ...
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40 views

How can a boundary measure of a function be absolutely continuous?

I'm studying firsts tools in several complex variables. In my book I found what follows: It can be proved that if $\varphi$ is strongly subharmonic and has a finite majorant in the unit ball, then ...
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105 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
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1answer
70 views

How do I show $\lim_{x\to\infty}f(x) = \lim_{x\to\infty} f '(x)=0$ if $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0$? [duplicate]

$f(x)$ is a real valued function on the reals, and has a continuous derivative such that $$\lim_{x\to\infty} f'(x)^2 + f(x)^3 = 0.$$ How do i show that $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} ...
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1answer
21 views

Determining continuity and differentiability

Is this function continuous and differentiable? $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x\:\ge 1 \end{array}\right.$$ For continuity, I did $$\lim_{x\to 1^+\:}f(x) = ...
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Analysis of continuity and differentiability of a function

Find a,b,c $\in \mathbb{R}$ for which the function is a) continuous, b) differentiable. $$f(x)=\left\{\begin{array}{cc} ax^2+bx+c & x<0 \\ 2\sin x+cos x & x\:\ge 0 \end{array}\right.$$ ...
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$f$ is monotone on D and $f(D)$ is an interval

$f$ is monotone on D and $f(D)$ is an interval then $f$ is continuous Is my proof right? pf) First, suppose it is monotone increasing Since $f(D)$ is an interval there is $[c,d]$ such that ...
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1answer
84 views

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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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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445 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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15 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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68 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
13 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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1answer
52 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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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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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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1answer
30 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
38 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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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 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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32 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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34 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
35 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
46 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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243 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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32 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 ...