For questions involving the concept of uniform continuity, that is, "the $\delta$ in the definition is independent of the considered point.

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19
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5answers
808 views

What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
0
votes
0answers
12 views

continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
1
vote
1answer
35 views

Proving that the characteristic function is uniformly continous.

I am trying to prove that the characteristic function is uniformly continuous. I understand how to get to this bound: And I would like to find the $\delta$ as a function of $\epsilon$ but I am ...
4
votes
2answers
67 views

“$\sigma$-uniform continuity”

Let $X$ be an arbitrary metric space and $f:X\to\mathbb R$ a bounded continuous function. Is it possible to choose a countable sequence $(A_n)_{n\in\mathbb N}$ of (preferably open or closed) subsets ...
2
votes
1answer
23 views

Surjectivity of expanding map

Suppose that $(X, d)$ is a compact metric space and that $f: X \rightarrow X$ is a continuous function satisfying $d(x,y) \leq d(f(x), f(y))$ for all $x, y \in X$. Show that $f(X) = X$. Here is a ...
2
votes
1answer
19 views

Expected loss in regards to a question containing a continuous random variable with uniform distribution

I have a general question about a homework assignment that deals with a uniform distribution of a continuous variable. Here is the question (and the parts of the question): Suppose parking rules are ...
0
votes
1answer
18 views

Show that if $g((x_n)) \rightarrow l$ and $g((y_n)) \rightarrow m$, then $l=m$

Suppose that $g: (a,b] \rightarrow \mathbb{R}$ is uniformly continuous. Suppose that both $(x_n), (y_n)$ are sequence in $(a,b]$ which converge to $a$. Show that if $g(x_n)) \rightarrow l$ and ...
0
votes
1answer
29 views

Show that $f$ is uniformly continuous.

Suppose that $F:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $x \rightarrow a$ of $f(x)$ exists. Show that $f$ is uniformly continuous. I am really struggling with this one. HELP ...
4
votes
1answer
44 views

prove that $\frac{1}{x}$ is not uniformly continuous on $(0,1)$

I would like to show that the function $\frac{1}{x}$ is not uniformly continuous on $(0,1)$ using two approaches. First Approach: We have the fact that if a function $f$ is uniformly continuous on ...
0
votes
0answers
20 views

How do I use Extreme Value Theorem and uniform continuity to prove this question?

I'm stuck on both parts of this question. Can somebody help me here?
0
votes
1answer
27 views

Prove this function is uniformly continuous by verifying the $\epsilon$-$\delta$ property?

$f(x) = \frac{5x}{2x-1}$ on $[1,\infty)$ Here's what I've worked through so far: $$|f(x) - f(y)| = \left|\frac{5x}{2x-1} - \frac{5y}{2y-1}\right| = \left|\frac{5y-5x}{(2x-1)(2y-1)}\right| ...
0
votes
2answers
40 views

Using only the definition of uniform continuity, prove that the following functions are uniform continuous:

Using only the definition of uniform continuity, prove that the following function is uniformly continuous: $g:[1,2] \to \mathbb R $ be defined by $g(x)=\sqrt x$ for all $x \in [1,2] $ Below is what ...
1
vote
1answer
32 views

Show that the metric space C[a,b] is complete. [duplicate]

Prove that the metric space $C[a,b]$ is complete. Where $C[a,b]$ is the collection of continuous $f:[a,b] → R$ and $||f|| = sup_{x \in [a,b]} |f(x)|$, such that $\rho (f,g) = ||f - g||$ is a metric ...
0
votes
1answer
52 views

Is the function $f: (1, \infty) \to \mathbb R$ defined as $f(x)=\sum_{n=1}^\infty n^{-x}$ continuous ?

Is the function $f: (1, \infty) \to \mathbb R$ defined as $f(x)=\sum_{n=1}^\infty n^{-x}$ continuous ? I know that for each $n \ge 1$ , the function $g:(1,\infty) \to \mathbb R ; g(x)=n^{-x}$ is ...
4
votes
1answer
43 views

Bounded Derivatives and Uniformly Continuous Functions

Prove or Disprove: Let $f:\mathbb{R} \to \mathbb{R}$ be a bounded uniformly continuous function that whose first and second derivative exists and is continuous, in other words $f \in C^2_{unif} ...
0
votes
1answer
40 views

Uniformly continuous on each of $A$ and $B$, not on $A\cup B$

I have a homework problem that asks me to find a function defined on subsets $A$ and $B$ of $\mathbb R$ such that the function is uniformly continuous on each of $A$ and $B$, and is continuous but not ...
0
votes
2answers
33 views

Show that the given function is a uniformly continuous function.

Let $F : \mathbb{R}^{n} → \mathbb{R}$ be defined by $F(x_1, x_2, . . . , x_n) = \max\{|x_1|, |x_2|, . . . , |x_n|\}$. Show that $F$ is a uniformly continuous function. I really have nothing to show ...
-1
votes
2answers
99 views

Find out where function is unfiormly continuous

Let $f(x)=\sin(\ln x)$ for $x>0$ find such $a,b,c,d>0$ where $f(x)$ is: uniformly continuous at intervals $(0,a], [b,+\infty)$ and Lipschitz at $(0,c]$ , $[d,+\infty)$
1
vote
1answer
40 views

Uniform continuity on $[c,d] \subseteq (a,b)$ implies UC on $(a,b)$ and UC on $[a,b] \ \forall a,b \in \mathbb{R}$ implies UC on $\mathbb{R}$

Is it true that if $f$ is uniformly continuous on $[c,d] \subseteq (a,b)$ then it is uniformly continuous on $(a,b)$? Furthermore, is it true that if $f$ is uniformly continuous on $[a,b] \ \forall ...
2
votes
1answer
56 views

If $f$ and $g$ are both uniformly continuous, show that $\max(f, g)$ is uniformly continuous

My friend asked me this question and I gave him a sketch of proof. My idea is that to construct a function $$h = \begin{cases} f-g & \textrm{if $f \ge g$}\\ 0 & \textrm{if $f < g$} ...
1
vote
0answers
17 views

Is the given function absolutely continuous

Define , $f:[0,1]\to \mathbb R$ by $$f(x)=\begin{cases}x\cos\frac{\pi}{2x}&\text{ for } x\not =0\\0&\text{ if } x=0\end{cases}$$ Then, $f$ is absolutely continuous or NOT ? I know that the ...
2
votes
1answer
50 views

Problem 8 (a) in Exercises after Sec. 18 in Munkres' Topology, 2nd ed.: How to show this set is closed? [duplicate]

Let $X$ be an arbitrary topological space, let $Y$ be an ordered set in the order topology, and let the maps $f, g \colon X \to Y$ be continuous. Then how to show that the set $S$ given by $$S \colon= ...
0
votes
1answer
20 views

Proving uniform continuity by definition

Here is the question: Use the definition of uniform continuity to prove $f(x)=x^2 + 2x - 5$ is uniformly continuous on $[0,3]$. I need help understanding how this works, and solving these types of ...
0
votes
1answer
28 views

Cauchy sequence under a uniform continuous function

Let , $f:(1,4)\to \mathbb R$ be uniformly continuous and $\{a_n\}$ be a Cauchy sequence in $(1,2)$. Consider: $x_n=a_n^2f(a_n^2)$ and $y_n=\frac{1}{1+a_n^2}f(a_n^2)$ Then which is ...
6
votes
2answers
75 views

Continuity and uniform continuity over $\mathbb Q$

Consider the function $f:\mathbb Q\to \mathbb R$ defined by $$f(x)=\begin{cases}1 &\text{ if, } x<\pi\\2 &\text{ if, } x>\pi\end{cases}$$ Show that $f$ is continuous but ...
2
votes
1answer
46 views

Show that the following are equivalent:

If $f$ is a continuous function on a bounded set $S$, show that the following are equivalent: (a) the function $f$ is uniformly continuous on $S$. (b) it is possible to extend $f$ to a ...
2
votes
3answers
69 views

Prove that $f(x) = \sqrt{x^2 + x}$ where $x \in [0, +\infty)$ is uniformly continuous

Prove that $f(x) = \sqrt{x^2 + x}$ where $x \in [0, +\infty)$ is uniformly continuous. So lets take: ${\mid \sqrt{x^2+x} - \sqrt{y^2+y} \mid}^2 \, \leqslant \,\, {\mid \sqrt{x^2+x} - \sqrt{y^2+y} ...
2
votes
3answers
78 views

Continuity at $+\infty$ for the function defined by $f(0)=\infty$ and $f(x)=1/x$ for $x \in (0,10]$.

Let the domain of the function $f(x)$ be $[0, 10]$, and its range be the extended real numbers (including +$\infty$ and -$\infty$). Define: $ f(x) = \left\{ \begin{array}{lr} 1/x & ...
0
votes
0answers
40 views

Inequality used in proof of continuity of Cantor function (a.k.a. Devil's staircase)

In my analysis course the lecturer constructs the Cantor function iteratively as is done here, then proves that the Cantor function is continuous as follows \begin{align} \max_{x \in [0, 1]} ...
5
votes
1answer
84 views

When does continuous (or uniformly continuous ) function between normed linear spaces carries bounded sets to bounded sets ?

I know that if $f:\mathbb R^m \to \mathbb R^n$ is continuous then $f$ carries bounded sets to bounded sets . What if we say $X,Y$ are normed linear spaces and $f:X \to Y$ where $f$ is continuous ? ...
1
vote
2answers
20 views

Continuity of an increasing function on a dense set

Let $f$ be increasing on $D$ ($D$ is dense in $\mathbb{R}$), and define $\tilde{f}$ on $(-\infty,\infty)$ as follows: $$ \forall x: \tilde{f}(x) = \inf_{x<t\in D} f(t).$$ Show that continuity of ...
2
votes
1answer
79 views

Boundedness of $f'(x)/x$ implies uniform continuity of $f(x)/x$ on $(1,\infty)$

Let $f:(1,\infty) \to \mathbb{R}$ be differentiable, define $g, h:(1,\infty) \to \mathbb{R}$ by $g(x)=f'(x)/x$ and $h(x)=f(x)/x$. Suppose $g$ is bounded. Prove that $h$ is uniformly continuous. I ...
0
votes
0answers
24 views

Why does this inequality imply a limit?

This is part of a proof taken from a textbook that a professor of my university wrote. The inequality is the following: $$0\le\sum_{i\in B}(M_i^*-m_i^*)\Delta_i\le2M\sum_{i\in ...
0
votes
0answers
27 views

Continuous to Uniform Continuous [duplicate]

Prove that if $f$ is continuous on $[a,\infty)$ and $\lim_{x\to\infty} f(x)=L$ exists, then $f$ is uniformly continuous on $[a,\infty)$. I know that I have to show that for all $\epsilon>0$ there ...
0
votes
1answer
34 views

Example of equivalent metrics on the same set such that uniform continuity of some function is not preserved

Give example of a set $X$ and two metrics $d_1,d_2$ on $X$ such that $(X,d_1)$ and $(X,d_2)$ are topologically equivalent but there exist a function $f:X \to X$ which is uniformly $d_1$ continuous but ...
3
votes
3answers
49 views

Need a help to show the following fuction is uniformly continuous.

$\newcommand{\dist}{\operatorname{dist}}$Suppose $ (X,\rho ) $ is a metric space and $ S $ is a non empty subset of $ X $. Then how to show the function $ g_S:X\rightarrow \mathbb{R} $ given by $ ...
2
votes
0answers
32 views

Set of polynomials of degree less or equal than $n$ is equicontinuous (or compact) over every interval $[a,b]$ using Arzela-Ascoli theorem

Define $\Pi=\{\text{polynomials of degree }\le n \text{ over } [a,b]\}$ with fixed $n$. Norm is $\|f\|=\sup_{x\in D(f)}|f(x)| $ I am trying to proof that this set is equicontinuous using ...
1
vote
1answer
44 views

Assuming continuity of $f$, how does one prove uniform continuity of $f$, assuming the limit to $+\infty$ and $-\infty$ is $0$?

Assuming continuity of $f$, how does one go about proving uniform continuity of $f$, assuming the limit to $+\infty$ and $-\infty$ is $0$? Note: $f\colon\mathbb R\to\mathbb R$. Looking for hints, ...
0
votes
0answers
42 views

Uniform continuity of a continuous function

Let $(X,d);(Y, \rho)$ be metric spaces , $(X,d)$ have nearest point property and $f:X \to Y$ is a continuous function ; then is it true that $f$ is uniformly continuous on any bounded set $A$ in $X$ ...
4
votes
1answer
44 views

Limit of naturals at infinity and uniform continuity

I need to prove the following: Let $ f:\mathbb{R}\to\mathbb{R}$ be continuous. $\lim_{n\to\infty}f(n) = \infty$, $n\in\mathbb{N}$. Let $f$ be uniformly continuous, prove that ...
0
votes
1answer
70 views

Let $f :\mathbb R\rightarrow\mathbb R$ be a function such that $f(x + 1) = f(x)$ for all x ∈ R. Which of the following statement(s) is/are true?

The given options are: (A) f is bounded. (B) f is bounded if it is continuous. (C) f is differentiable if it is continuous. (D) f is uniformly continuous if it is continuous. Any hints on how to ...
0
votes
1answer
80 views

Prove that a function is not globally Lipschitz

Here is the function : $f(t,x)=e^{tx}$ on $\mathbb{R\times R}$. First the function is $\mathcal{C^1}$ so it's locally Lipschtiz in $x$ on $\mathbb{R\times R}$. Then I try to prove if it's globally ...
8
votes
1answer
141 views

Uniformly continuous maps between topological groups

Let $G$ be a topological group. For every neighbourhood $U$ of the identity, let $L_U$ be the set of all pairs $(x,y) \in G \times G$ such that $x^{-1} y \in U$. For topological groups $G$ and $H$, a ...
2
votes
2answers
63 views

Is $f(x)=\frac{\cos \left(x\right)}{1+x^2}$ uniformly continuous?

Consider $f:[0,\infty)\rightarrow \mathbb{R}$:$$f(x)=\frac{\cos \left(x\right)}{1+x^2}$$Is $f$ is uniformly continuous on $[0,\infty)$? So far I tried to bound the derivative of this, but i get: ...
3
votes
3answers
63 views

Example 4.21 in Baby Rudin: How is the map $f^{-1}$ not continuous at the point $(1,0) = f(0)$?

Let $f \colon [0,2\pi ) \to \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\}$ be defined as $$f(t) = ( \cos t , \sin t) \ \ \mbox{ for all } \ t \in [0, 2\pi).$$ Then the map $f$ is bijective and ...
1
vote
1answer
44 views

Theorem 4.20(c) in Baby Rudin: Is every continuous function whose domain is an unbounded subset of $\mathbb{R}$ uniformly continuous?

Here is Theorem 4.20 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Let $E$ be a non-compact set in $\mathbb{R}^1$. Then (a) there exists a continuous function on ...
1
vote
2answers
92 views

Theorem 4.20 in Baby Rudin: How is this map not uniformly continuous?

Let $E$ be a bounded, non-compact subset of $\mathbb{R}$, let $x_0$ be a limit point of $E$ such that $x_0 \not\in E$, and let $f \colon E \to \mathbb{R}$ be defined by $$f(x) \colon= \frac{1}{x-x_0} ...
0
votes
1answer
20 views

Show that $m(\Gamma)=0$, where $\Gamma$ is a curve $y=f(x)$

Suppose $\Gamma$ is a curve $y=f(x)$ in $\mathbb{R}^2$, where $f$ is continuous. Show that $m(\Gamma)=0$. Hint: Cover $\Gamma$ by rectangles, using the uniform continuity of $f$. If the ...
1
vote
1answer
45 views

Proving uniform continuity of sequence of functions

Lets take the infinite sequence of functions $f_{n}(x) = x/(x +1/n) , x \in [0, 1], n \in \Bbb N$. Show that each function $f_{n}$ is uniformly continuous. My solution: Given $\epsilon >0$, let ...
0
votes
0answers
22 views

Find parameters so that $f(x)$ be uniformly continuous or lipschitz continuous

Let $f(x)=\sin(\ln(x))$ for $x>0$ $1)$ Find every $a,b>0$ so that $f(x)$ is uniformly continuous at intervals $(0,a] , [b,+\infty)$ $2)$ Find every $c,d>0$ so that $f(x)$ is lipschitz ...