Question

Is every monotonic additive function $f \colon \mathbb{R} \to \mathbb{R}$ continuous?

score 421 · Answer 1 · 2026-03-25 12:55:14

421

Views

Is every monotonic additive function $f \colon \mathbb{R} \to \mathbb{R}$ continuous?

Published on 25 Mar 2026 - 12:55

score 121 · Answer 2 · 2026-03-26 06:33:51

121

Views

Proof that the following function is continuous and (hence holomorphic using Morera's).

Published on 26 Mar 2026 - 6:33

score 41 · Answer 3 · 2026-03-28 22:55:20

41

Views

Uniform convergence of logarithm on a compact set.

Published on 28 Mar 2026 - 22:55

score 416 · Answer 4 · 2018-09-05 13:40:47

416

Views

Justify if $fg$ is differentiable at point $x_0=0$

Published on 05 Sep 2018 - 13:40

score 145 · Answer 5 · 2026-03-26 06:31:54

145

Views

Proof continuity of following function: $g(z)=\int_{\gamma} \frac{f(z)-f(w)}{z-w}dw$

Published on 26 Mar 2026 - 6:31

score 852 · Answer 6 · 2026-03-25 18:50:13

852

Views

Show that there is a continuous real function $h$ on $[0,1]$ such that $\lim \sup |\frac{h(x+t)-h(x)}{t}| = \infty$ for all $x \in [0,1)$

Published on 25 Mar 2026 - 18:50

score 59 · Answer 7 · 2018-09-06 07:11:46

59

Views

Prove that the function $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=x^\frac{1}{n}$ is continuous

Published on 06 Sep 2018 - 7:11

score 287 · Answer 8 · 2026-03-25 08:10:47

287

Views

Show that the projection from $C[0,1]$ onto $P_n[0,1]$ under $L^q$ norm is nonlinear when $q>1$ and $q\neq 2$.

Published on 25 Mar 2026 - 8:10

score 229 · Answer 9 · 2026-03-25 14:24:33

229

Views

Definition 5.6.6 in Bartle & Sherbert's INTRO TO REAL ANALYSIS: How to show that this notion is well-defined?

Published on 25 Mar 2026 - 14:24

score 40 · Answer 10 · 2018-09-06 12:23:08

$f: [a,b] \to \Bbb R$ is continous on $[a,b]$ then $f([a,b])=[m,M]$ s.t $f(a) \neq m$ , $f(a) \neq M$ then $ \exists c \in (a,b)$ s.t $f(c)=f(a)$.

score 221 · Answer 11 · 2018-09-06 17:02:56

221

Views

No. of solutions of $f(x)=f'(x)$?

Published on 06 Sep 2018 - 17:02

score 209 · Answer 12 · 2026-04-02 20:58:22

209

Views

How to connect the definitions of open sets and continuous functions w.r.t. metric space and topological space?

Published on 02 Apr 2026 - 20:58

score 4k · Answer 13 · 2018-09-06 23:10:48

4k

Views

What does it mean when $f(a)$ is defined?

Published on 06 Sep 2018 - 23:10

score 206 · Answer 14 · 2018-09-07 05:21:14

206

Views

If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$

Published on 07 Sep 2018 - 5:21

score 329 · Answer 15 · 2026-03-25 16:03:52

329

Views

Prob. 6, Sec. 5.6, in Bartle & Sherbert's INTRO TO REAL ANALYSIS: A necessary and sufficient condition for continuity of a monotonic function

Published on 25 Mar 2026 - 16:03

List Question

Is every monotonic additive function $f \colon \mathbb{R} \to \mathbb{R}$ continuous?

Proof that the following function is continuous and (hence holomorphic using Morera's).

Uniform convergence of logarithm on a compact set.

Justify if $fg$ is differentiable at point $x_0=0$

Proof continuity of following function: $g(z)=\int_{\gamma} \frac{f(z)-f(w)}{z-w}dw$

Show that there is a continuous real function $h$ on $[0,1]$ such that $\lim \sup |\frac{h(x+t)-h(x)}{t}| = \infty$ for all $x \in [0,1)$

Prove that the function $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=x^\frac{1}{n}$ is continuous

Show that the projection from $C[0,1]$ onto $P_n[0,1]$ under $L^q$ norm is nonlinear when $q>1$ and $q\neq 2$.

Definition 5.6.6 in Bartle & Sherbert's INTRO TO REAL ANALYSIS: How to show that this notion is well-defined?

$f: [a,b] \to \Bbb R$ is continous on $[a,b]$ then $f([a,b])=[m,M]$ s.t $f(a) \neq m$ , $f(a) \neq M$ then $ \exists c \in (a,b)$ s.t $f(c)=f(a)$.

No. of solutions of $f(x)=f'(x)$?

How to connect the definitions of open sets and continuous functions w.r.t. metric space and topological space?

What does it mean when $f(a)$ is defined?

If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$

Prob. 6, Sec. 5.6, in Bartle & Sherbert's INTRO TO REAL ANALYSIS: A necessary and sufficient condition for continuity of a monotonic function

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