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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7
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2answers
112 views

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
2
votes
1answer
32 views

differentiable and uniform continuity of f and F

Given $f: \Bbb R \to \Bbb R$. define new function: $F(x) =\frac{f(x)-f(a)}{x-a}$ for $x\neq a$. Prove that $f$ is differentiable at $a$ if and only if $F$ is uniformly continuous in some punctured ...
3
votes
1answer
35 views

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent:

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent: a) $f$ is uniformly continuous in $X$. b)For every pair of sequences $(x_n), (y_n) \subseteq X$ such that $ d(x_n, ...
3
votes
2answers
46 views

Continuity vs. Mapping open sets to open sets?

I have a question and I have no idea how to solve this: One problem in my Real Analysis text book says: Show that if $\ell$ is a nonzero linear functional on a normed vector space not necessarily ...
1
vote
1answer
23 views

inverse of uniformly continuous function is uniformly continuous?

Inverse of uniformly continuous function is uniformly continuous? Assume that $ X,Y$ are metric spaces and let $f:X\to Y$ such that $f$ is bijective and ...
2
votes
2answers
36 views

Axiom of choice : continuous function and uniformly continuous

How I proof that every continuous function f in [0,1] is uniformly continuous, without axiom o choice? I took this from the book Axiom of Choice from Horst Herrilich He had a observation that ...
1
vote
1answer
44 views

If $f: (a, b) \to \Bbb R$ is uniformly continuous, then $g: [a, b] \to \Bbb R$ is continuous

Fact 1: If $A$ is a subset of a metric space, $Y$ is a metric space, and $f: A \to Y$ is uniformly continuous, then if $\{x_n\}$ is Cauchy in $A$, then $\{f(x_n)\}$ is Cauchy in $Y$. Fact 2: ...
1
vote
1answer
25 views

Uniform continuity preserves uniqueness of convergent sequences?

If $f: (a, b) \to \Bbb R$ is uniformly continuous, $\{x_n\}$ and $\{x'_n\}$ are sequences in $(a, b)$ with $x_n \to b$, $x'_n \to b$, $f(x_n) \to y$, and $f(x'_n) \to \overline{y}$, prove $y = y'$. ...
0
votes
0answers
30 views

Proving continuity using Weierstrass M-test

I am asked to use the Weierstrass M-test to show that the following function is continuous on $A = \mathbb{R}\setminus \mathbb{Z}$ $$f(x) = \sum_{n=1}^\infty \frac{1}{x+n} + \frac{1}{x-n}$$ My ...
1
vote
3answers
102 views

Is $\frac{1}{x}$ on [0,$\infty$] continuous at zero?

Taking the definition of continuity, two of three conditions are met, i.e. a) We would have to define $f(0)=\infty$, but normally division by zero is not well-defined. b) The limit $\lim_{x\to ...
39
votes
3answers
2k views

Why did mathematicians introduce the concept of uniform continuity?

I have solved many problems regarding uniform continuity, but still I can't understand the following: Is there any practical application of this concept, or it is just a theoretical concept? Is there ...
0
votes
0answers
46 views

Is this function bounded and what about uniform continuity?

There is a function $ \phi (x): \mathbb{R}^n \longmapsto \mathbb{R}^n$ which satisfies conditions (1) and (2). (1): $\langle\phi(x_1)-\phi(x_2),x_1-x_2\rangle \leq \gamma_1 \| x_1-x_2 \|^2$ (2): $ ...
1
vote
2answers
50 views

Show $f(x) = x\sin{x}$ is not uniformly continuous on $\mathbb{R}$ [duplicate]

I know it's not Lipschitz continuous because its derivative is unbounded but I'm not sure if not Lipschitz continuous implies not uniformly continuous. Thanks
3
votes
1answer
30 views

Difference between continuous and uniformly continuous functions on a dense metric subspace.

Let $X$ be a dense subset of metric space $(\tilde X,d)$. Let be $(Y,d')$ be a complete metric space and $ f: X \rightarrow Y$ a continuous mapping. It follows from density that for all points in ...
0
votes
2answers
77 views

Is $\sqrt{x}$ uniformly continous in $\mathbb{R}^+$?

We are given this function: $f:R^+\rightarrow R,x\rightarrow \sqrt{x}$. We need to prove that this function is uniformly continuous. My proof is this one but i'm not sure is it complete and right. ...
0
votes
1answer
26 views

Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that \begin{equation} |u(x)-u(y)| \leq ...
23
votes
5answers
1k 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
14 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
37 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
68 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
30 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
35 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
20 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 ...
1
vote
2answers
47 views

Show that $f$ is uniformly continuous.

Suppose that $f:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $\lim\limits_{x \rightarrow a}f(x)$ exists. Show that $f$ is uniformly continuous. I am really struggling with this ...
4
votes
1answer
50 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
22 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
31 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
42 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
41 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
55 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
48 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
42 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
34 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
100 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
41 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
62 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
22 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
56 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= ...
-1
votes
1answer
23 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
32 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
97 views

Continuity and uniform continuity of $f(x)$ 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
49 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
76 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
79 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
41 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
96 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
22 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
81 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 ...