Question

Let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Suppose that $E$ is a bounded subset of $X$. Then $f(E)$ is also bounded.

score 123 · Answer 1 · 2026-03-26 12:38:01

123

Views

Let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Suppose that $E$ is a bounded subset of $X$. Then $f(E)$ is also bounded.

Published on 26 Mar 2026 - 12:38

score 69 · Answer 2 · 2020-05-01 18:09:21

69

Views

How do we gain a better intuition on the definition of uniform continuity and its advantages compared to usual continuity?

Published on 01 May 2020 - 18:09

score 577 · Answer 3 · 2020-05-01 18:35:25

577

Views

Prove that the composition of uniformly continuous functions is uniformly continuous.

Published on 01 May 2020 - 18:35

score 226 · Answer 4 · 2026-03-28 21:56:58

226

Views

Cauchy sequence of a uniformly continuous image

Published on 28 Mar 2026 - 21:56

score 98 · Answer 5 · 2020-05-02 08:46:45

98

Views

Is "bounded" and "Cauchy-continuous" function uniformly continuous?

Published on 02 May 2020 - 8:46

score 211 · Answer 6 · 2020-05-02 11:31:11

211

Views

$f>0$ uniformly continuous and $\int_0^{\infty} f(x) \,dx \leq M$ imply $\lim_{x \to \infty} f(x)=0$?

Published on 02 May 2020 - 11:31

score 340 · Answer 7 · 2020-05-02 16:00:03

340

Views

Continuity implies uniform continuity

Published on 02 May 2020 - 16:00

score 119 · Answer 8 · 2020-05-03 04:54:57

119

Views

How to prove the uniform equivalence is indeed an equivalence relation on the class of metrics on X

Published on 03 May 2020 - 4:54

score 35 · Answer 9 · 2020-05-03 06:55:19

35

Views

Some questions about the proof of the properties of the uniform equivalent.

Published on 03 May 2020 - 6:55

score 41 · Answer 10 · 2020-05-03 08:16:50

41

Views

Example of the two uniformly equivalent metrics, one is bounded while another is not.

Published on 03 May 2020 - 8:16

score 155 · Answer 11 · 2020-05-03 12:12:30

155

Views

$g(x) = f(x)\sin\ (1/x)$ being uniformly continuous on $(0, 1]$

Published on 03 May 2020 - 12:12

score 45 · Answer 12 · 2020-05-03 22:29:34

45

Views

Proving non-uniform continuity of functions.

Published on 03 May 2020 - 22:29

score 52 · Answer 13 · 2026-02-23 04:59:33

52

Views

Uniform convergence - Is f continuous and is f differentiable $f(x) = \sum_{n=1}^{\infty}\frac{1}{2n^2-\sin(nx)}$

Published on 23 Feb 2026 - 4:59

score 68 · Answer 14 · 2020-05-13 02:19:00

68

Views

Prove that $f:X\rightarrow\Bbb{R}$ is uniformly continuous iff it maps equivalent sequences onto equivalent sequences

Published on 13 May 2020 - 2:19

score 296 · Answer 15 · 2026-02-24 07:03:12

296

Views

Uniform Continuity of a Piecewise Function

Published on 24 Feb 2026 - 7:03

List Question

Let $f:X\rightarrow\textbf{R}$ be a uniformly continuous function. Suppose that $E$ is a bounded subset of $X$. Then $f(E)$ is also bounded.

How do we gain a better intuition on the definition of uniform continuity and its advantages compared to usual continuity?

Prove that the composition of uniformly continuous functions is uniformly continuous.

Cauchy sequence of a uniformly continuous image

Is "bounded" and "Cauchy-continuous" function uniformly continuous?

$f>0$ uniformly continuous and $\int_0^{\infty} f(x) \,dx \leq M$ imply $\lim_{x \to \infty} f(x)=0$?

Continuity implies uniform continuity

How to prove the uniform equivalence is indeed an equivalence relation on the class of metrics on X

Some questions about the proof of the properties of the uniform equivalent.

Example of the two uniformly equivalent metrics, one is bounded while another is not.

$g(x) = f(x)\sin\ (1/x)$ being uniformly continuous on $(0, 1]$

Proving non-uniform continuity of functions.

Uniform convergence - Is f continuous and is f differentiable $f(x) = \sum_{n=1}^{\infty}\frac{1}{2n^2-\sin(nx)}$

Prove that $f:X\rightarrow\Bbb{R}$ is uniformly continuous iff it maps equivalent sequences onto equivalent sequences

Uniform Continuity of a Piecewise Function

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