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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3
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1answer
72 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
0
votes
1answer
17 views

Prove that if f is an element of Lip a and a>1 then f is constant

We have that a function $f: [a,b]\mapsto\Bbb R$ satisfies a Lipschitz condition of order $a > 0$ if there is some positive constant $M$ so that $$|f(x_1)-f(x_2)| \leq M|x_1-x_2|^a \qquad \forall ...
0
votes
1answer
28 views

Prove that if f is an element of Lip a, then f is uniformly continuous

I am required to prove that if f is an element of Lip a, then f is uniformly continuous. Where Lip means a Lipschitz function. I also have to prove that if f is an element of Lip a and a>1 then f is ...
0
votes
2answers
80 views

Show that $g(x)=\sqrt{x}$ is uniformly continuous on $[0, \infty)$

I must show that $g(x)=\sqrt{x}$ is uniformly continuous on $[0, \infty)$. I know we have to show that $\sqrt{a-b}\ge \sqrt{a} - \sqrt{b}$ and $\sqrt{a + b} \le \sqrt{a} + \sqrt{b}$ but I don't know ...
0
votes
1answer
22 views

Uniform continuity of the function: [on hold]

Test the uniform continuity of the function $f(x)=x^{2/3}\log x$ where $x$ belongs to $(0, \infty)$.
0
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1answer
22 views

Conditions for open interval continuity

Please can someone help in giving me the condition that would make a continuous function on an open interval be uniformly continuous in that same interval. Thanks.
0
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1answer
18 views

Uniform Continuous function f has limit zero at infinity

Suppose f(x) if uniformly continuous in $[0,+\infty)$. Prove that if $\{f(x+n)\}_{n}$ converges to $0$ $\forall x \in [0,1]$, then $\lim_{x->\infty} f(x) = 0$.
1
vote
0answers
28 views

Multivariate Weierstrass theorem?

The Weierstrass theorem states that for any continuous function $f$ of one variable there is a sequence of polynomials that uniformly converge to $f$. To my surprise, I couldn't find any reference to ...
0
votes
3answers
54 views

Show that if $f$ is continuous on $(0,1)$ and $\lim_{x \to 0}f(x) = + \infty$ then $f$ is not uniformly continuous

I must show that if $f$ is continuous on $(0,1)$ and $$\lim_{x \to 0}f(x) = + \infty$$ then $f$ is not uniformly continuous. I am not sure if we have to use epsilon- delta proofs or how to show this.
1
vote
1answer
37 views

Prove that an Interval I is closed and bounded.

I'm attempting to prove the following statement: Let $I$ be a non-trivial interval over the real numbers. Show that if every continuous function on $I$ is uniformly continuous, then $I$ is closed ...
1
vote
1answer
64 views

Prove that $f(x)=x\sin(x)$ is not a uniformly continuous function

I have this problem, and im really lost here, i need to show that. $f(x)=x\sin x$ is not an uniformly continuous function.
1
vote
1answer
40 views

Proving that the function f is of class C^1,

Suppose $f:R->R$ is continuous, and that it has a continuous right derivative, i.e. the right-sided limit $$lim(\delta->0^+) (f(x+\delta)-f(x))/\delta$$ exists for all x $\in$ R and defines a ...
1
vote
1answer
86 views

Prove that the function $f(x) = \sin(1/x)$ is not uniformly continuous

Prove that the function defined by $f(x) = \sin(1/x)$ is not uniformly continuous on the interval $(0,1)$. Hint: Consider for example $x = 1/2nπ$ and $y = 1/[(2n+1/2)π]$ I have an answer with let $ε ...
2
votes
1answer
40 views

Vector-valued function, proving whether it's continuous, based on its action on any line in R^2:

Suppose $f: R^2 -> R^2$ is a function whose restriction to any line L in $R^2$ is continuous. Prove or find a counterexample: f must be continuous. For starters, I drew an arbitrary point on the ...
0
votes
1answer
22 views

Riemann Integrable — Is this true?

I have a sequence of functions $f_n$ (on a compact interval) which uniformly converge to $f$, and I know $f$ is Riemann Integrable. Is it true that $f_n$ must also be Riemann Integrable?
2
votes
1answer
28 views

Proof of uniformly continuous for nth root of x on [0,+∞)

In my lecture notes I saw the proof of $f(x) = \sqrt x$ is uniformly continous on $[0,+∞)$. The proof goes as follows: Given $\epsilon >0$, we pick $\delta = \epsilon ^2$. We note that $|\sqrt ...
0
votes
1answer
11 views

sequential criterion

This is one of the practice questions I'm working on: Use the Sequential Criterion to prove that $f(x) = \sin(x^2)$ is not uniformly continuous on $\Bbb R$. I need some help to get started, for ...
2
votes
1answer
29 views

Show that f is uniformly continuous on [0, +∞)

So working on an exercise from my notes, I am given the conditions that $f$ is continuous on $[0, +∞)$ and uniformly continuous on $[a,+∞)$ for some $a > 0$. How do I show that $f$ is uniformly ...
2
votes
4answers
356 views

What does continuity *in general* mean?

I am looking from : http://en.wikipedia.org/wiki/Lipschitz_continuity Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ α-Hölder continuous $\subseteq$ uniformly continuous ...
2
votes
0answers
24 views

A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ...
1
vote
5answers
90 views

If $f$ is uniformly continuous on $(a, b)$, then $f$ is bounded on $(a, b)$.

So I know that since f is uniformly continuous on (a, b), then for every $\epsilon > 0$, there exists $\delta > 0$ such that for all x and y in (a, b), if |x - y| < delta, then |f(x) - f(y)| ...
0
votes
1answer
47 views

Say $f : \mathbb{R} \to \mathbb{R}$ is continuous and $f(x) \to 0$ as $x \to \pm\infty$. Show $f$ is uniformly continuous.

I have a problem from Carother's Real Analysis, page 116. Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous and $f(x) \to 0$ as $x \to \pm\infty$. Prove that $f$ is uniformly continuous. ...
-1
votes
2answers
29 views

$\lim_{x \to + \infty} f(x)$ when $f$ is uniformly continuous

Does the $\lim_{x \to + \infty}$ f(x) of an uniformly continuous function exist? How to show it? Applying the definition of uniform continuity I cannot show that...
2
votes
2answers
35 views

Help with details in an example showing the function $f(x)=\frac{1}{x^2}$ is not uniformly continuous

Here is a portion of an example in Ross' Elementary Analysis. Definition 19.1 is the definition of uniform continuity. What is the rationale for choosing $\frac{\delta}{2}$?
2
votes
2answers
58 views

Help showing the distance function is uniformly continuous.

Let $X$ be a metric space and $q \in X$. I want to show that the distance function $d(q,p)$ is a uniformly continuous function of $p$. I know how to show that $d$ is continuous, but I am stuck on ...
1
vote
1answer
32 views

uniformly continuous when second derivative is bounded

Let $f$ be continuous on $[a,b]$. $f$ is twice differentiable on $(a,b)$ and $|f^{\prime\prime}(x)|\leq M$ for all $x\in [a,b]$. Show $f$ is uniformly continuous. This is a question from an exam in ...
0
votes
1answer
41 views

Finding an unbounded set with a specific property

Find an unbounded subset $A ⊂ \Bbb R$ such that every function from $A$ to a metric space is uniformly continuous. My attempt at the solution (incomplete). If $A⊆ \Bbb R$ were such a set, then for ...
0
votes
1answer
41 views

An Application of Intermediate Value Theorem

Let $f :\Bbb R→ \Bbb R$ be given by $f(x) := x^{n}$ for some $n ∈ \Bbb N$. If $b$ is a positive real number, show that there exists a unique positive real number $a$ such that $a^{n} = b$. My ...
1
vote
1answer
49 views

Real analysis. Uniformly continuous

Suppose $$f:\mathbb R\to\mathbb C$$ is continuous and $f(x)=0$ for all $|x|>1$. Show $f$ is uniformly continuous on $\mathbb R$. This is not homework. I'm trying to study for a test. I ...
1
vote
1answer
15 views

Proving uniform continuity using limits

Hi I am interested in a result which states that if a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ has finite limits on both sides then the function is uniformly continuous. Is it ...
4
votes
3answers
37 views

Every function from discrete metric space to another metric space is uniformly continuous

My solution:It is fairly straightforward graphically but I just want to ensure if it is rigorous enough. Suppose $X$ is a discrete metric space and $f$ be any function from $X$ to $Y$ where $Y$ is ...
4
votes
1answer
49 views

A closed, bounded subset $A$ of $\Bbb Q$ and a continuous function $f : A → \Bbb R$ such that $f$ is not bounded

Find a closed, bounded subset $A$ of $\Bbb Q$ and a continuous function $f : A →\Bbb R$ such that $f$ is not bounded Note: $\Bbb Q$ is the set of all rationals. My Solution: $A=\{x:x\in\Bbb Q, ...
0
votes
2answers
34 views

If $f$ is a uniformly continous function, then $|f(x)|\leq a|x|+b$

Suppose $f$ is a uniformly continous function. Prove that there exists $a,b\in\mathbb{R}$ such that for any $x$: $$|f(x)|\leq a|x|+b$$ I proved it for a Lipschitz function with constant $k$ and ...
1
vote
1answer
23 views

Existence of Limit iff $x',x'' > X, |f(x')-f(x'')| < \epsilon$

I was given a theorem in class regarding uniform continuity that does not appear in my textbook. It says that $$\lim_{x \to \infty} f(x) = a \iff \text{ for all } x',x'' > X, |f(x')-f(x'')| ...
0
votes
1answer
23 views

Uniform Continuity on two separate intervals, but no on the union of the intervals

I'm just trying to find a function that is uniformly continuous on sets D1 and D2, but not on the union of D1 and D2. Thank you!
2
votes
1answer
53 views

Uniform limit of uniformly continuous functions

Let $f_n:\mathbb{R}\to\mathbb{R}$ be a sequence of uniformly continuous functions. Assume that $f_n$ converges uniformly on all bounded intervals $[a,b]$ to a function $f$, i.e. ...
1
vote
1answer
27 views

A calculus problem regarding continuity and uniform continuity

A function $f : \mathbb{R} \to \mathbb{R} $ is continuous at the origin $x=0$ and we have $$ f(0) = 0 $$ $$ \forall x_1 \in \mathbb{R} ,\forall x_2 \in \mathbb{R}: f(x_1 + x_2 ) \leq f(x_1) + ...
11
votes
1answer
199 views

Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?

Spending the night perusing my old answers, and this question left me wondering about the following. Let's equip $\Bbb{R}^n$ with the usual Euclidean metric, and let us consider the map ...
0
votes
1answer
119 views

Given I is bounded. Show if f: I → R is uniformly continuous on I, then f is bounded on I.

a) Let I be a bounded interval. Prove that if f: I → R is uniformly continuous on I, then f is bounded on I. Proof: Suppose I is a bounded interval, such that I = (a,b). Then given f: (a,b) → R, by ...
4
votes
4answers
92 views

Show $1/(1+ x^2)$ is uniformly continuous on $\Bbb R$.

Prove $\frac 1{1+ x^2}$ is uniformly continuous on $\Bbb R$. Attempt: By definition a function $f: E →\Bbb R$ is uniformly continuous iff for every $ε > 0$, there is a $δ > 0$ such that $|x-a| ...
4
votes
1answer
45 views

Prove that f is uniformly continuous on [0, ∞].

Exercise: Suppose that f:[0, ∞] → R is continuous, and that there is an real value L, such that f(x) → L as x → ∞. Prove that f is uniformly continuous on [0, ∞]. Attempt: I will try to use the ...
1
vote
0answers
18 views

Is the following function uniformly continuous?

I am supposed to prove if the function $ f:X \to \mathbb {R}, f (x) = dist (x, A)$ where $ A$ is an arbitrary subset o the metric space $X $ is uniformly continuous. If both points $ x $ and $ y $ ...
1
vote
1answer
23 views

If a function is continuous on $\mathbb R,$ does it follow that it is uniformly continuous on $(-1,1)?$

I've been trying to think of counterexamples but the ideas I've had so far like $1/(x+1)$ and $\sin(1/(x+1))$ don't work because those aren't continuous on all of $\mathbb R.$
2
votes
1answer
29 views

Show the function is uniformly continouos on (0,1).

Prove that each of the following functions is uniformly continuous on (0,1). (You may use l'Hopital's Rule). b)f(x) = xcos(1/x^2) attemp in proof: I need to use the following theorem. Theorem 3.40: ...
0
votes
1answer
27 views

Is $A$ compact, $f(A)$ uniformly continuous and is $f^{-1}$ continuous?

$X$ and $Y$ are metric spaces, $A\subseteq X$, $A$ is bounded. map $f:X\to Y$ is continuous. Questions: Is $A$ necessarily compact? Is $f(A)$ uniformly continuous? If given that $f$ is a ...
0
votes
1answer
29 views

Using unbounded derivative to show function is not uniformly convergent

I'm confused how to use the following theorem: 19.6 Theorem. Let $f$ be a continuous function on an interval $I$ [$I$ may be bounded or unbounded]. Let $I^◦$ be the interval obtained by removing ...
0
votes
2answers
71 views

The product of uniformly continuous functions is not necessarily uniformly continuous

I was asked to show that given two functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ which are both uniformly continuous, to show that the product ...
0
votes
0answers
16 views

A bounded set in a hilbert space with a compact domain is equicontinuous?

I came across this line in a book "As bounded sets in $H^1(\Omega)$ are equi-continuous and $\Omega$ is compact..." It goes on to prove a result from this but what has me stuck is I don't see is ...
0
votes
2answers
68 views

Which of the following is true about $f(x)$?

If $f(x)=x+\sin x$, then which of the following is true about $f(x)$? $1.f(x)$ is uniformly continuous on $\mathbb{R}$. $2.f(x)$ has bounded variation on $\mathbb{R}$. $3.f(x)$ does not have ...
0
votes
1answer
20 views

Will uniform continuity of $g$ imply the uniform continuity of $f$?

Let $S\subset \mathbb R$ and $g:S\rightarrow \mathbb R$ be uniformly continuous. If $f:S\rightarrow \mathbb R$ be a continuous function such that $f(x)\leq g(x)\forall x\in S$, can we conclude that ...