Question

Prove that if $W$ is an open subset of $Y$ then the set $\{x\in X: f(x)\in W\}$ is an open subset of $X$.

score 162 · Answer 1 · 2018-05-11 18:05:15

162

Views

Prove that if $W$ is an open subset of $Y$ then the set $\{x\in X: f(x)\in W\}$ is an open subset of $X$.

Published on 11 May 2018 - 18:05

score 2.2k · Answer 2 · 2026-03-25 04:55:36

2.2k

Views

Mobius band is differentiable manifold

Published on 25 Mar 2026 - 4:55

score 31 · Answer 3 · 2018-05-12 01:46:54

31

Views

inequality of two decreasing functions

Published on 12 May 2018 - 1:46

score 250 · Answer 4 · 2026-03-26 03:12:23

250

Views

Proof writing: Let $f:[a,b]\to \Bbb{R}$ be continuous. Then $f$ is uniformly continuous

Published on 26 Mar 2026 - 3:12

score 62 · Answer 5 · 2026-03-25 21:51:11

62

Views

Show that if $X:[0,T]\to\mathbb R$ is continuous, then $\inf\left\{t\in\overline I:|X(t)|\geε\right\}\le t$ iff $|X(s)|\geε$ for some $s\in[0,t]$

Published on 25 Mar 2026 - 21:51

score 358 · Answer 6 · 2026-03-26 03:13:33

358

Views

$E$ and $F$ Banach Spaces, $T:E\to F$ such that $\text{Im }T$ is closed. Then $\text{Im } T^{\ast }=\text{ann }\left ( \ker T \right )$

Published on 26 Mar 2026 - 3:13

score 162 · Answer 7 · 2026-03-25 21:48:58

162

Views

Continuity for a set of functions in Supremum Norm metric space

Published on 25 Mar 2026 - 21:48

score 47 · Answer 8 · 2026-03-26 04:53:12

47

Views

If $E$ is a Banach space and $S\subset E^{\ast}$ is $w^{\ast}$-closed, then $S=\text{ann }(S_0)$ for some $S_0\subset E$

Published on 26 Mar 2026 - 4:53

score 106 · Answer 9 · 2018-05-12 22:03:51

106

Views

Continuity of $x^\alpha$

Published on 12 May 2018 - 22:03

score 236 · Answer 10 · 2018-05-13 00:21:26

236

Views

Proving $\cos(x+y)+\sin(xy)=x^2+y$ has infinite solutions using intermediate value theorem

Published on 13 May 2018 - 0:21

score 115 · Answer 11 · 2026-03-26 04:53:08

115

Views

Prove that f(x) vanishes in whole of its domain.

Published on 26 Mar 2026 - 4:53

score 925 · Answer 12 · 2026-03-25 15:57:09

925

Views

Shrinking map theorem

Published on 25 Mar 2026 - 15:57

score 197 · Answer 13 · 2018-05-13 18:12:01

197

Views

Continuity and adherence in topology

Published on 13 May 2018 - 18:12

score 4.5k · Answer 14 · 2018-05-13 20:06:44

4.5k

Views

Prove that $f(x)=\begin{cases}1& \text{if x is rational},\\0 &\text{if x is irrational}\end{cases}$ is discontinuous at every real number.

Published on 13 May 2018 - 20:06

score 79 · Answer 15 · 2018-05-14 02:37:41

79

Views

Prove the following equation has infinitely many solutions and the set of solutions is a closed set

Published on 14 May 2018 - 2:37

List Question

Prove that if $W$ is an open subset of $Y$ then the set $\{x\in X: f(x)\in W\}$ is an open subset of $X$.

Mobius band is differentiable manifold

inequality of two decreasing functions

Proof writing: Let $f:[a,b]\to \Bbb{R}$ be continuous. Then $f$ is uniformly continuous

Show that if $X:[0,T]\to\mathbb R$ is continuous, then $\inf\left\{t\in\overline I:|X(t)|\geε\right\}\le t$ iff $|X(s)|\geε$ for some $s\in[0,t]$

$E$ and $F$ Banach Spaces, $T:E\to F$ such that $\text{Im }T$ is closed. Then $\text{Im } T^{\ast }=\text{ann }\left ( \ker T \right )$

Continuity for a set of functions in Supremum Norm metric space

If $E$ is a Banach space and $S\subset E^{\ast}$ is $w^{\ast}$-closed, then $S=\text{ann }(S_0)$ for some $S_0\subset E$

Continuity of $x^\alpha$

Proving $\cos(x+y)+\sin(xy)=x^2+y$ has infinite solutions using intermediate value theorem

Prove that f(x) vanishes in whole of its domain.

Shrinking map theorem

Continuity and adherence in topology

Prove that $f(x)=\begin{cases}1& \text{if x is rational},\\0 &\text{if x is irrational}\end{cases}$ is discontinuous at every real number.

Prove the following equation has infinitely many solutions and the set of solutions is a closed set

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