Question

Homeomorphism: can $f:X\mapsto X'$ be continuous, bijective without $f^{-1}:X'\mapsto X$ being continuous?

score 166 · Answer 1 · 2019-01-20 10:00:17

166

Views

Homeomorphism: can $f:X\mapsto X'$ be continuous, bijective without $f^{-1}:X'\mapsto X$ being continuous?

Published on 20 Jan 2019 - 10:00

score 1.5k · Answer 2 · 2026-03-25 22:09:18

1.5k

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Composition of discontinuous functions

Published on 25 Mar 2026 - 22:09

score 651 · Answer 3 · 2019-01-20 13:40:09

Proof for continuity: $f (x) = \left\{ \begin{array}{ll} \frac{2xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & \, (x,y)=(0,0) \\ \end{array} \right. $

score 31 · Answer 4 · 2019-01-20 14:24:21

31

Views

Determine the limit of the function $f$ at $0$, for the function defined on $\mathbb{R}$ by $f(0)=0$ and $f(x)=\sin(1/x)$ for $x \neq 0$.

Published on 20 Jan 2019 - 14:24

score 166 · Answer 5 · 2019-01-20 15:10:52

166

Views

Does Intermediate value theorem work depending on the domain

Published on 20 Jan 2019 - 15:10

score 92 · Answer 6 · 2019-01-20 18:37:40

92

Views

Extending a function defined in $\mathbb{R}^n\setminus\{0\}$ to a continuous function defined in $\mathbb{R}^{n}$.

Published on 20 Jan 2019 - 18:37

score 272 · Answer 7 · 2019-01-21 15:06:34

272

Views

What is the meaning of the statement $2<x<10$?

Published on 21 Jan 2019 - 15:06

score 242 · Answer 8 · 2026-03-30 15:27:48

242

Views

Continuous function on compact metric space has minimum and maximum

Published on 30 Mar 2026 - 15:27

score 228 · Answer 9 · 2026-03-25 21:46:52

228

Views

Inverse function of a product space

Published on 25 Mar 2026 - 21:46

score 45 · Answer 10 · 2026-03-26 14:43:16

45

Views

Continuity of Delta in Delta-Epsilon Argument

Published on 26 Mar 2026 - 14:43

score 1.2k · Answer 11 · 2019-01-22 02:46:13

1.2k

Views

A function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is open and closed, but not continuous.

Published on 22 Jan 2019 - 2:46

score 1k · Answer 12 · 2019-01-22 04:50:37

1k

Views

In which of the following cases is there no continuous function from the set $S$ onto the set $T$?

Published on 22 Jan 2019 - 4:50

score 143 · Answer 13 · 2026-03-27 06:17:01

143

Views

$f$ is continuous function on $[0, \infty)$, and the limit $\lim_{n \to \infty} \frac{f(x)}{x}= a \in \mathbb{R}$ exists. $f$ is uniformly continuous?

Published on 27 Mar 2026 - 6:17

score 367 · Answer 14 · 2019-01-22 17:03:00

367

Views

Continuity of homotopy in proof of Hopf's Umlaufsatz

Published on 22 Jan 2019 - 17:03

score 238 · Answer 15 · 2019-01-22 19:11:00

238

Views

Prove $f(\overline{U})) \subset \overline{f(U)}$

Published on 22 Jan 2019 - 19:11

List Question

Homeomorphism: can $f:X\mapsto X'$ be continuous, bijective without $f^{-1}:X'\mapsto X$ being continuous?

Composition of discontinuous functions

Proof for continuity: $f (x) = \left\{ \begin{array}{ll} \frac{2xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & \, (x,y)=(0,0) \\ \end{array} \right. $

Determine the limit of the function $f$ at $0$, for the function defined on $\mathbb{R}$ by $f(0)=0$ and $f(x)=\sin(1/x)$ for $x \neq 0$.

Does Intermediate value theorem work depending on the domain

Extending a function defined in $\mathbb{R}^n\setminus\{0\}$ to a continuous function defined in $\mathbb{R}^{n}$.

What is the meaning of the statement $2<x<10$?

Continuous function on compact metric space has minimum and maximum

Inverse function of a product space

Continuity of Delta in Delta-Epsilon Argument

A function $f:\mathbb{R}^2\to\mathbb{R}^2$ that is open and closed, but not continuous.

In which of the following cases is there no continuous function from the set $S$ onto the set $T$?

$f$ is continuous function on $[0, \infty)$, and the limit $\lim_{n \to \infty} \frac{f(x)}{x}= a \in \mathbb{R}$ exists. $f$ is uniformly continuous?

Continuity of homotopy in proof of Hopf's Umlaufsatz

Prove $f(\overline{U})) \subset \overline{f(U)}$

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