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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A question on continuity of a piecewise function with 4 constants

I have this function, and I need to find the values of $a, b, c$ and $d$ so that $f(x)$ will be differentiable everywhere. $$f(x)=\begin{cases} ax+b, & x<-2 \\ x^2+c, & -2\le x\le2\\ ...
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1answer
46 views

Show that the sequence does not converge

My Try: $|f'(a)|>1$. Assume that the sequence converges to a limit $b$. Then $f(b)=b$. Since $a$ is the only fixed point it implies that $b=a$. Hence, given any $\frac{1}{m}$ where $m\in ...
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4answers
78 views

Proving that a continuous $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A) \text{ connected}$

Proving that $f:X \to Y ; \ X,Y- \text{topological spaces}$ and $A \subseteq X ; A \text{ connected} \implies f(A)-\text{connected}$ The answer is given like this just one step I do not understand ...
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2answers
86 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
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1answer
29 views

Question about continuity of piecewise function of two variables

Let $$ f(x,y)= \left\{ \begin{array}{ll} \left(x\sin\left(\frac{y}{x}\right),\frac{\cos (y) -1}{y}\right) & x \neq 0 \wedge y \neq 0 \\ (0,0) & x = 0 \vee y = 0 \\ \end{array} ...
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2answers
34 views

Piecewise $\mathscr C^1$ and piecewise continuous

I'm a little bit confused in piecewise continuity of a function. Say, if we have an odd function like $f(x) = x$ defined on the open interval $(0, \pi)$. We then extend it to a period $2\pi$ function ...
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2answers
141 views

continuity of a function

I have a task as preparation for my Calculus Exam. $f(x)= \begin{cases} 2^{\frac{1}{x-2}} ,& x\neq 2 \\ 0 ,&x=2 \end{cases}$ Now we have the following solution by one of our tutors: $l_1 = ...
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1answer
23 views

Showing that a continuous function is greater than zero

I've been working on a problem that wants me to show that given a function $f$ that is continuous at the point $c$ that, $$f(c)>0 \to \exists \delta\;\ \text{such that}\;\ f(x)>0\; \forall x ...
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1answer
40 views

Proving that is $A:X \implies Y$ is a linear operator from metric space X to Y is continuous iff it is bounded bounded

The $\implies$ part interests me. The proof given goes like this: Let $A$ be continuous in 0 (because the 0 vector is in every vector space) $B_y(0,r)=\{y \in Y | \| y\|<r \} \implies \exists ...
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17 views

is this multivariable function twice continuously differentiable with respect to the parameter?

I have the following function $V: {R}^{*}_{+} \times {R}^{*}_{+} \rightarrow R$ , a and b are strictly positive real coefficients: $$V ( x_i(l) , x_j (l) ; l ) = a x_i (l) - b x_i (l)^2 + l^2 x_i ...
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1answer
47 views

Show continuity using epsilon delta definition for piecewise function [closed]

Using epsilon delta definition, show that $g$ is continuous on the whole of $\mathbb R$ $$g(x)=\cases{x^2 & \text{ if } x<1\\ \sqrt{x} & \text{ if } x≥1.}$$
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1answer
61 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
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0answers
35 views

Continuity by composition with a homeomorphism

I only want to know what do you guys think about the following proof. That's an exercise I've tried to do and I don't have an available answer, so... If you find some error or imprecision, I'd be ...
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0answers
16 views

continuous random variable - pth percentile

Let X be a loss random variable with cdf $$ F(x) = \left\{ \begin{array}{ll} 1-(θ/θ+x)^α & \textrm{for $x≥0$}\\ 0 & \textrm{for $x<0$}\\ \end{array} \right. $$ The 10th percentile is θ−k. ...
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1answer
47 views

Proof of the continuity of a function at irrational points

The problem is to prove that, If $f:\mathbb{R}\to\mathbb{R}$ defined by $$ \begin{align} f(x) = \begin{cases} 0 & \text{if $x\in \mathbb{R}\setminus\mathbb{Q}$}\\ \dfrac{1}{n} & \text{if ...
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1answer
24 views

(Just for clarification) - Is a convex, piecewise continuous function f on an closed interval continuous?

Lets say f is defined on an Interval $I = [a,b] $. Since f is convex, one immediately knows that f is continuous on $I^°$ , however left are the points $a$ and $b$ The piecewise continuity of f ...
2
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1answer
24 views

Prove continuity for a given norm

I struggle with this exercise from an analysis 2 book I use for self study: Let V := $C^1([0,1]; \mathbb{C})$ the vector space of continously differentiable functions from $[0,1]$ to $\mathbb{C}$ ...
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1answer
84 views

Is there continuous $f: [0, 1] \rightarrow [0, \infty)$ such that for all $x$ there is $y$ with $f(y) < f(x)$?

I think there isn't. Here's a sketch of a proof. I'm just not sure whether it really works because I'm not confident with the transfinite versions of the standard theorems about limits and convergent ...
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2answers
54 views

Why the length of the zigzag curve approximating the circle does not approach the length of the circle?

I recently bumped into this question which asks why $\pi=4$ is wrong. And some answers(see the answer of user TCL, for example) stated that this has to do with functions and their derivatives. ...
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4answers
363 views

Why do Topologies get “finer”?

Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that ...
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1answer
32 views

Prove continuity of averaging function for integrable $f$

I want to prove the following statement which is part of a lemma in my textbook: Suppose $f$ is integrable on $\mathbb{R}^n$ and $x$ be a lebesgue point of $f$. Let $$M(r)=\frac{1}{r^d}\int_{|y|\le ...
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1answer
25 views

Evaluation function is Lipschitz wrt uniform conv metric

In the book on Brownian motion by Schilling and Praetzsch there is following statement: Let $\mathcal{C}_{(0)}:=\{f\in\mathcal{C}[0,\infty):\ f(0)=0\}$ be the space of all continuous functions ...
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1answer
69 views

Continuity of a function on $\Bbb R^2$ [closed]

Function $f(x,y)$ is defined in a neighborhood of $(0,0)$. Then if for any t function $g(x) = f(x,tx)$ is continuous at $0$, then $f$ is continuous at $(0,0)$. if $f$ is continuous at ...
6
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1answer
94 views

Continuous map $\mathbb{R}^n\rightarrow\mathbb{R}^n$

When we say some map $\phi=(\phi_1,\ldots,\phi_n)$ is a continuous map $\mathbb{R}^n\rightarrow\mathbb{R}^n$ we really mean that each component $\phi_i$ is continuous as a function ...
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1answer
77 views

Need to prove continuous periodic function of $\varphi (x) \equiv \psi(x)$

Question: Let two $\varphi(x) $ and $\psi(x)$ periodic and continous functions such that $$ \lim_{ x\to\infty}(\varphi(x)-\psi(x))=0, \quad x\in \mathbb{R}. $$ Prove that $$ ...
3
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2answers
73 views

Integrating over a somewhat continuous function

I have a function $q(t)$ that starts at $q(0)=q_0$ and needs to get to $q(1)=q_1>q_0$. I have a free parameter $z$ that I can wiggle around to control $q'(t)$. Namely, I have a function ...
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1answer
59 views

Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?

Is it true that for every compact subset $A$ of $\mathbb R^2$ , there exist a compact set $B$ in $\mathbb R$ such that there is a continuous surjection from $B$ to $A$ ?
3
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3answers
107 views

Which statement “must be false”?

Given a function $f$ continuous on $[-4, 1]$ with its maximum at $(-3, 5)$ and its minimum at $(1/2, -6)$, is it not correct to say that both statements (B) and (D) must be false? (A) The graph of ...
4
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1answer
66 views

Show that $f(x) = \cos(2x)$ is uniformly continuous on $[0,\infty)$

Let $f: [0,\infty) \to \mathbb{R}$ and let $f(x) = \cos(2x)$. Show that $f(x)$ is uniformly continuous on $[0,\infty)$ Mt attempt: We have, $\forall \epsilon >0, \exists \delta > 0, s.t.\mid ...
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1answer
55 views

Continuity at $x=0$ of this function

Not a hard exercise:$$f(x)=\frac{1}{x^3}\cdot \int_{-x}^x \sin(4t^2) \, \text{d}t \quad \text{where} \space x\ne 0\:$$ $$f(x)=5\:;\:x=0\:$$ Checking it's continuity at $x=0$ by using L'Hospital's ...
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1answer
57 views

2 exercises: finding the limit and showing continuity and differentiability

part 1: $$\lim _{x\to _{x\to \frac{\pi }{2}^{-\:\:}} }\left(tg\left(x\right)\right)^{\sin\left(2x\right)}$$ so if $$\lim _{x\to _{x\to \frac{\pi }{2}^{-\:\:}} ...
3
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2answers
60 views

Prove that $f_A (x) = d({\{x}\}, A)$, is continuous.

Prove that: Let $(X, d)$ be a metric space, and let $A$ be a subset of $X$. The function $f_A\colon X\rightarrow \mathbb{R}$, defined by $f_A (x) = d({\{x}\}, A)$, is continuous. Honestly, I ...
2
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1answer
25 views

Help with proof; Pre-image of a continuous function around a maximum is open.

Suppose a function $u$ is defined in an open and connected set $D$ and has maximum value $c$. Then if $u$ is not constant in D then the set $\{u(z) < c \mid z \in D\}$ is non-empty and open. This ...
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4answers
71 views

Why isn't the initial topology always the trivial topology?

If I have a set $X$ and a function $f:X\rightarrow X$, then I think $f$ is continuous with the trivial topology, because no matter what the function is, $f(X)\subseteq X$. Thus for any point $f(x)$, ...
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2answers
37 views

Limit of a function with 2 variables

I am given this function: $$f(x,y)=\begin{cases}\frac{xy^3}{x^2+y^4} & \text{ for } (x,y)\not=(0,0)\\ 0 & \text{ for } (x,y)=(0,0)\end{cases}$$ and I have to check if it is continuous in ...
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1answer
23 views

Convergence in probability of a composite function.

Question: Let $f$ be a continuous function on $\mathbb{R}.$ If $X_n \to X$ in probability, then $f(X_n) \to f(X)$ in probability. The result is false if $f$ is merely Borel measurable. [Hint: ...
3
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2answers
58 views

Proving characterization of continuity with direct images of sets using nets.

We know that if $X,Y$ are topological spaces, then $f: X \to Y$ is continuous if and only if $f(\overline{E}) \subseteq \overline{f(E)}$, for all $E \subseteq X$. I started studying nets by myself ...
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5answers
275 views

Is sine of one degree a real? If not, how is sine continuous?

If I understand it correctly, the impossibility of trisection of an arbitrary angle implies that sine of one degree isn't a real number, but how is it then possible for sine to be continuous, if it ...
3
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2answers
36 views

Proving that $(X,\tau)$ is Hausdorff given a condition.

Let $(X,\tau)$ be a topological space such that for each $p \in X$ there is a continuous function $f:X \to \Bbb R$ verifying $f^{-1}(\{0\}) = \{p\}$. Then $(X,\tau)$ is Hausdorff. Welp, take $p,q ...
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2answers
45 views

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 ...
3
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1answer
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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2answers
47 views

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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1answer
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 ...
0
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2answers
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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3answers
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 ...
2
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0answers
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 ...
2
votes
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?
1
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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 ...
1
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2answers
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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2answers
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?