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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Discontinuous for rationals

Show that $f\left(x\right):=\sum_{n=1}^{\infty}\frac{\left\{nx\right\}}{n^2}$, where $\left\{nx\right\}$ is the fractional part of $nx$, is discontinuous for all rationals. I guess it would be nice ...
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0answers
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Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
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1answer
13 views

A continuity question

Find a non-zero value for the constant k that makes $f(x)=\begin{Bmatrix} \dfrac{\tan(kx)}{x} ,& x<0 \\[6pt] 3x+2k^{2}, & x\geqslant 0 \end{Bmatrix}$ continous at $x=0$. I've been trying ...
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2answers
33 views

Intermediate value theorem problem

Problem: The equation $x=-5\cos(x)$ has at least $3$ distinction solutions. Use the intermediate value theorem to show that this is true. I drew the function,but I don't know what to do next.
3
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1answer
37 views

Continuity in $\mathbb R^n$.

we just got started with this topic today, and I am confused. Let $f:\Bbb R^2 \to \Bbb R $ with $$f(x,y) =\begin{cases} y\sin(x)/x &\text{if } x \ne 0\\ 0 &\text{else} \end{cases}$$ Now, ...
0
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1answer
20 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
4
votes
1answer
77 views

Are “most” continuous functions also differentiable?

Let $A$ be a nonempty open subset of $\mathbb{R}$. Consider a function $f : A \rightarrow \mathbb{R}$. Given that $f$ is continuous, what is the probability that it is differentiable? I suspect it ...
1
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1answer
13 views

need some help with this continuity

Find a non-zero value for the constant k that makes $f(x)=\begin{Bmatrix} \dfrac{\tan(kx)}{x} ,& x<0 \\[6pt] 3x+2k^{2}, & x\geqslant 0 \end{Bmatrix}$ continous at $x=0$. I tried to do this ...
0
votes
1answer
18 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
1
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1answer
32 views

Isomorphism between rings

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$. 1) Prove that $I$ is an ideal. ...
2
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1answer
57 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
3
votes
1answer
27 views

Relation between continuity of $f$ and analyticity of $f(z)^8$

If $f(z)$ is continuous on some domain $D$ and $f(z)^8$ (the function to the eighth power, not the eighth derivative) is analytic, then why does this imply that f is analytic on a neighborhood of each ...
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1answer
28 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
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1answer
25 views

Piece-Wise Function

Give an example of a function $f$ whose domain is the closed interval $[0,1]$ such that $f$ is bounded but does not attain its upper bound (i.e. there is no $x_1$ that exists in $[0, 1]$ such that ...
1
vote
1answer
28 views

Intermediate Value Theorem help

Let $f$ be a continuous function on $\mathbb{R}$ which is periodic with period $2\pi$. This means $f(t + 2\pi) = f(t)$ for all $t$. Show that there exists $x\in[0,\pi]$ such that $f(x) = f(x + \pi)$. ...
29
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9answers
727 views

A game with $\delta$, $\epsilon$ and uniform continuity.

UPDATE: Bounty awarded, but it is still shady about what f) is. In Makarov's Selected Problems in Real Analysis there's this challenging problem: Describe the set of functions $f: \mathbb R ...
4
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2answers
34 views

For $f$ a continuous topological mapping, when are the values on the boundary of a set determined?

Suppose $f:X\to Y$ is a continuous map between topological spaces, and suppose we know the value of $f$ on a subset $S\subset X$. Continuity tells us that $f(\bar{S})\subset \overline{f(S)}$ for any ...
1
vote
1answer
35 views

If $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$

If a mapping $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$ The mapping $f$ is injective as $f(x) = ...
0
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1answer
37 views

Help, check the uniform continuity

(1) $f(x)=sin(1/x)$ on $(0,1]$ ? ( I know it is not uniform continuous on $(0,1)$) (2) $f(x)= xsin(1/x)$ on $(0,1]$? (3) $f(x)=sin(x^2)$ on $[0, \infty)$?
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0answers
22 views

Some Continuity Question

Suppose $f(x)$ and $g(x)$ are continuous functions on $[a,b]$ with $f$ monotone increasing. Assume there exists a sequence $x_n \in [a, b]$ such that for all $n \in \mathbb{N}$ , $g(x_n) = ...
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0answers
25 views

upper hemicontinuity

Let $g: \mathbb R^2_+ \to \mathbb R_+$ and $h: \mathbb R^2_+ \to \mathbb R_+$ continous functions. For every $ t \in \mathbb R_+$, 1) $g(t, \cdot)$ has a unique maximum at $V(t)$ where $V: \mathbb ...
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1answer
22 views

Continuity Function Problem

Suppose f(x) is a continuous function from [0,1] into [0,1]. Show that there exists a point $\xi \in [0,1]$ such that $f(\xi) = \xi$.
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3answers
45 views

Show a function is not continuous

let $g(x) = x - \lfloor{x}\rfloor$ and I want to show that the function is not continuous. I want to use this definition im pretty sure: "For every open set U in $R$, $f^{-1}$ U is open" But I am ...
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1answer
1k views

Multi-variable continuity piecewise problem

I've worked on this for about 3 hours and I can't seem to get anywhere with it. I tried using java code to return the solution but one that met the criteria was not found. Find the value of $c$ and ...
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2answers
46 views

Why is this subset not open?

I have a function, $f:[0, 1) \rightarrow \mathbb{S}^1$ given by $f(x) = (\cos2\pi x, \sin2\pi x)$. I have to show that $f$ is bijective and continuous and that $f^{-1}$ is not continuous. I have ...
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1answer
24 views

Questions on Continuous Function

I know that it is very obvious that intuitively, a continuous function cannot have any gap in between. However, I am having difficulty proving it. Normally, in textbook and also in my real analysis, ...
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0answers
10 views

Functions in a Reproducing Kernel Hilbert Space are Lipschitz continuous

I would like to show that all the functions in a Reproducing Kernel Hilbert Space (RKHS) are Lipschitz continuous. So that, I take two points in the domain $\vec{x}_{1} ,\vec{x}_{2} \in X$ then from ...
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0answers
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Zoo of sigmoid integrals (computational convenience)

In many areas in computational science (e.g. neural networks, fuzzy logic ... ) there is special interest in function like sigmoid ( erf, arctan, tanh ... ) which are kind of blured version of ...
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0answers
37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
1
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1answer
18 views

continuous functions on metric space

Assume $f:X\rightarrow Y$, where $X$ and $Y$ are two metric spaces. If $f(\overline{E})\subset \overline{f(E)}, \, \forall E\subset X$, then how can we prove that $f$ is continuous? Thank you for ...
3
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0answers
24 views

Surjective function on a compact metric space [duplicate]

Assume $f:K\rightarrow K$, is surjective and $K$ is a compact metric space and we have $d(f(x),f(y))\leq d(x,y)\, \forall x,y\in K$. How can I prove that $d(f(x),f(y))= d(x,y)\, \forall x,y\in K$? ...
1
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1answer
16 views

How to prove $f(x)=e^{\frac{1}{x}}$ is continous in $(0,a), a>0 $ and $\int_{0}^{a}e^{\frac{y}{x}}dx, y>0$ does not exist

I would aprecciate any advice. I'm trying to prove that in the context of a measure space, $(X,B,\lambda)$ , with $X=(0, + \infty) $, $B$ the Borel sigma-algebra and $\lambda$ the Lebesgue measure, ...
0
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2answers
26 views

Continuity of an operator in $C^0[0,1]$ with different norm

Let $C^0[0,1]$ be the space of real valued continuos functions with the norm $\|f\| = \int \limits_{0}^1 x^2 |f(x)| dx$ and let $T \colon C^0[0,1] \to C^0[0,1]$ such that $f(x) \mapsto f(1-x)$. Is $T$ ...
0
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0answers
14 views

Continuity of Complex function and restrictions

I am trying the following question but am stuck at finding the restriction: Prove that $f(z)=1/z^2$ is continuous at $z_0= 1+2i$ Solution: I am trying the use epsilon-delta proof and got it down to: ...
0
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1answer
22 views

Discontinuity of the indicator function

Consider the function $q(x,\theta)=1\{ x \in \{x \text{ s.t. } \theta+x_i>0 \text{ }\forall i \}\}$ where 1 is the indicator function taking value 1 if the condition inside $\{ \}$ is satisfied and ...
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votes
1answer
32 views

Some questions about uniform continuity

I got the following questions: Let $f$ be a real valued function of a real variable: (1) If $f$ is continuous and bounded on the interval (a,b) (meaning there exist $M,L\in \mathbb{R}$ such that ...
0
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0answers
17 views

Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
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votes
1answer
38 views

A basic confusion over uniform continuity

Suppose $F$ defined on $[a,b]$ is continuous. Is this true that $$ \sup_{0 < h < \frac{1}{n}} \frac{F(x+h) - F(x)}{h} \leq \sup_{h \in \text{rationals between 0 and 1/n}} \frac{F(x+h) - ...
0
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2answers
29 views

If f is continuos on an interval, is it then uniformly continuous

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I now know that it is not. Can someone give me a proof ...
0
votes
1answer
23 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...
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votes
1answer
60 views

Countablity of the set of the points where the characteristic function of the Cantor set is not continous

We are creating the Cantor set typically starting from the interval $[0,1]$ and removing $\frac{1}{3}$ of it like it is described here or here. The problem is to resolve if the set of discontinuities ...
0
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0answers
16 views

A question about a function that is uniformly continuous and nonzero on an interval

I got this question: Let $f$ be a function defined on an interval $I$ and let $0<L$ be a constant, If $f$ is uniformly continuous on $I$ and $\forall x\in I, L \leq f(x)$, Must it be the case that ...
0
votes
2answers
17 views

An MCQ question on continuity.

Let f: R->R be a continuous bounded function, then : A. f has to be uniform continuous. B. There exists an x in R such that f(x)=x C. f cannot be increasing. D. lim x->inf f(x) exists. A ...
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0answers
10 views

Uniform law of large numbers for discontinuous functions?

Do you know about any Uniform Law of Large numbers (see http://en.wikipedia.org/wiki/Law_of_large_numbers#Uniform_law_of_large_numbers) that work when f is the indicator function (and thus not ...
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0answers
21 views

Is convergence in probability to a uniformly continuous function a sufficient condition for stochastic equicontinuity?

Suppose that a random function $g_T(\theta)$ converges in probability to a function $g(\theta)$ uniformly continuous over $\Theta$ as $T\rightarrow \infty$ $\forall \theta \in \Theta$. Is this ...
0
votes
1answer
27 views

A question about continuous functions from a closed interval into itself

I got this question: Let $f, g:[a,b]\to [a,b]$ be functions that are continuous on [a,b] such that $g$ is onto $[a,b]$, Must it be the case that $\exists x \in [a,b]$ such that $f(x) = g(x)$? This ...
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1answer
23 views

A question about the relation between supremums of two functions that satisfy certain properties

I got this question: Let $f, g : [a,b] \to [a,b]$ be functions that satisfy $\forall x \in [a,b], g(x) < f(x)$. Prove or give a counter example for the following statements: (1) If $f$ and $g$ ...
0
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0answers
10 views

Let C(R) be all real valued continuous functions on R such that limx->+-infinity f(x)=0 show that C(R) is complete with respect to the uniform metric.

I have looked through many other proofs online but none seem to provide a general proof for this they all seem specific to the interval [0,1] and those that are not do not seem to explicitly prove it. ...
0
votes
1answer
11 views

Composite Continuous Function

Let $g$ and $h$ be real-valued functions with domains $\operatorname{dom}(g)$ and $\operatorname{dom}(h)$ respectively. Suppose that $g$ maps $\operatorname{dom}(g)$ into $\operatorname{dom}(h)$, that ...
0
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1answer
23 views

Continuity Cubic Function [closed]

let $c$ be the cubing function $c(u) = u^3$. verify the identity $c(u)-c(v) = (u-v)(u^2 + uv + v^2)$ and use it to prove the function $c$ is continuous at each point of $\mathbb{R}$.