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) ...

learn more… | top users | synonyms

0
votes
0answers
8 views

With $\omega(\delta)$ being the modulus of continuity, prove $\omega( \delta_1 + \delta_2) \leq \omega(\delta_1) + \omega(\delta_2)$

If the modulus of continuity for the function $f: E \to \Bbb R$ is the function $\omega(\delta)$ defined for $\delta > 0$ by $$\omega(\delta) = \sup_{|x_1 - x_2| < \delta}_{x_1, x_2 \in \Bbb E} ...
1
vote
1answer
26 views

Continuity of bilinear maps

Given a vector space $V$ over $\mathbb{R}$ with a norm $||*|| $. Can $(x,y)\rightarrow(x+y)$ be an example of continous bilinear map, if yes, can you please exlain why? Definition of continuous ...
0
votes
1answer
17 views

Metric induced from norm

I was trying to understand the following: Every norm on $R^n$ is continuous (as a map from $R^n$ to $R$). Proof. We use the maximum metric on $R^n$: $ d(x, y) = \max{|x_j − y_j| : j ∈ \{1, . . . ...
1
vote
1answer
19 views

Definition of upper hemicontinuity of a correspondence.

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...
1
vote
3answers
17 views

Negation of continuity applied to a sequence

Show that if it is not true that $\lim_{x \to a} f(x)=l$ then $\exists$ $\epsilon$>0 and a sequence $(x_{n}) \rightarrow a$ as $n \rightarrow \infty$ such that $|f(x_{n})-l| \geq \epsilon$. Now ...
0
votes
2answers
23 views

Continuous Function on closed interval

I am having trouble understanding what this question is asking , by "$f$ has a zero" does it mean "there exists $x$ $\in$ $[a,b]$ such that $f(x)$=$0$? any help on how to answer this question in both ...
1
vote
1answer
31 views

A question about a continuous function that satisfy certain limits at $\pm\infty$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $\lim_{x\to\infty}\frac{f(x)}{x^2}$ and $\lim_{x\to -\infty}\frac{f(x)}{x^2}$ exist and are real numbers. ...
0
votes
3answers
15 views

Find the real parameters for which the function is continuous over the reals.

How should I solve this exercise? Find the values of the real parameters $a$ and $b$ for which the following function is continuous on $\Bbb{R}$: \begin{cases} e^x+a\cos(x) \text{ if } x\le0 \\ ...
1
vote
1answer
29 views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ that got no extrema must be one to one

I got this question: Prove that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function that got no extrema then $f$ is one to one. I tried to prove it but I don't know how to proceed. I started by ...
-1
votes
0answers
40 views

Study the continuity of a function [on hold]

Determine if the following function is continuous $$f(x) = \lim_{n\to\infty}\frac{x}{(1+2\sin x)^{2n}}$$
0
votes
2answers
29 views

Continuity of 1/x

I am confused with what $8(ii)$ wants from me, I answered the first part of this question with help from the question posted here Is $f(x)=1/x$ continuous on $(0,\infty)$? But the this proves ...
0
votes
1answer
22 views

Proving Uniform Continuity using Bolzano Weierstrass

I have been working on this question for some times, and can't seem to put together the contradiction needed using Bolzano. any help would be greatly appreciated,
0
votes
1answer
15 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 ...
3
votes
1answer
39 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
votes
2answers
34 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.
0
votes
1answer
22 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 ...
1
vote
1answer
15 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 ...
4
votes
1answer
102 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
vote
1answer
33 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. ...
6
votes
0answers
36 views

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 $Y$ ...
0
votes
1answer
22 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
vote
1answer
30 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
2
votes
1answer
59 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 ...
0
votes
1answer
26 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
29 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 $f(x) = f(x + \pi)$. ...
4
votes
2answers
37 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
38 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
votes
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 $[0, \infty)$?
0
votes
0answers
23 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) = ...
0
votes
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 $f(\xi) = \xi$.
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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, ...
1
vote
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 ...
1
vote
0answers
16 views

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 ...
0
votes
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
vote
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
votes
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
vote
1answer
18 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
votes
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
votes
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
votes
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 ...
-2
votes
1answer
35 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
votes
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?
2
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
votes
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
24 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 ...
0
votes
0answers
18 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 ...