Question

$1\leq p<\infty$ and $f\in L^p(R)$. $g(x)=\int_x^{x+1} f(t)dt$. Show that $g$ is continuous. What if $f\in L^{\infty} (\mathbb{R})$?

score 239 · Answer 1 · 2026-04-16 13:54:00

239

Views

$1\leq p<\infty$ and $f\in L^p(R)$. $g(x)=\int_x^{x+1} f(t)dt$. Show that $g$ is continuous. What if $f\in L^{\infty} (\mathbb{R})$?

Published on 16 Apr 2026 - 13:54

score 144 · Answer 2 · 2026-04-15 18:01:54

144

Views

Prove an equivalent definition of $\sup\{T>0:x\in\overline{\mathbb R^n\setminus D_t },t \in [0,T) \},D_t\subset\mathbb R^n,x\in\partial D_0$

Published on 15 Apr 2026 - 18:01

score 262 · Answer 3 · 2026-05-10 19:16:41

262

Views

Equibounded $L_1[0,1]$ is not compact metric space with cardinality $2^\mathbb{R}$?

Published on 10 May 2026 - 19:16

score 172 · Answer 4 · 2026-05-10 15:03:18

172

Views

Evaluating Riemann Summation Limit: $\lim\limits_{n\to \infty}\sum\limits_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}}$

Published on 10 May 2026 - 15:03

score 82 · Answer 5 · 2026-04-16 07:53:03

82

Views

Given $0<\lambda<1.$ Prove that if all $x,y>0,$ $f(x+y) = \lambda f(x) + (1-\lambda)f(y),$ then $f$ is a constant function.

Published on 16 Apr 2026 - 7:53

score 154 · Answer 6 · 2026-05-14 15:00:22

154

Views

Fundamental Theorem of Calculus . Why is this wrong

Published on 14 May 2026 - 15:00

score 740 · Answer 7 · 2026-04-24 09:41:52

740

Views

show that the recursive sequence converges to $\sqrt r$

Published on 24 Apr 2026 - 9:41

score 37 · Answer 8 · 2026-05-14 22:07:39

37

Views

A question regarding connected subsets of connected metric spaces

Published on 14 May 2026 - 22:07

score 559 · Answer 9 · 2026-05-10 12:33:25

559

Views

Let$ f : [0,1] \to [0,1] $ be a strictly increasing function. Show that it has a fixed point.

Published on 10 May 2026 - 12:33

score 45 · Answer 10 · 2026-04-14 19:45:05

45

Views

Given $\sum\limits_{k=1}^\infty \gamma_k r_k<\infty$ and $\gamma_k\not\in\ell^1$, then $\lim\inf r_k=0$

Published on 14 Apr 2026 - 19:45

score 63 · Answer 11 · 2026-04-24 11:17:19

63

Views

Is this sequence converging monotonically?

Published on 24 Apr 2026 - 11:17

score 121 · Answer 12 · 2026-05-14 13:52:31

121

Views

Which of the following of subspaces of $\mathbb{R^2}$ are compact?

Published on 14 May 2026 - 13:52

score 31 · Answer 13 · 2026-04-16 13:03:35

31

Views

A real function sending bounded sets to bounded sets is necessarily of bounded variation?

Published on 16 Apr 2026 - 13:03

score 460 · Answer 14 · 2026-04-22 11:23:30

460

Views

Continuity of composition of two-variables and one-variable functions

Published on 22 Apr 2026 - 11:23

score 9.3k · Answer 15 · 2026-05-13 05:25:12

9.3k

Views

Orthogonality and norm of Hermite polynomials

Published on 13 May 2026 - 5:25

List Question

$1\leq p<\infty$ and $f\in L^p(R)$. $g(x)=\int_x^{x+1} f(t)dt$. Show that $g$ is continuous. What if $f\in L^{\infty} (\mathbb{R})$?

Prove an equivalent definition of $\sup\{T>0:x\in\overline{\mathbb R^n\setminus D_t },t \in [0,T) \},D_t\subset\mathbb R^n,x\in\partial D_0$

Equibounded $L_1[0,1]$ is not compact metric space with cardinality $2^\mathbb{R}$?

Evaluating Riemann Summation Limit: $\lim\limits_{n\to \infty}\sum\limits_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}}$

Given $0<\lambda<1.$ Prove that if all $x,y>0,$ $f(x+y) = \lambda f(x) + (1-\lambda)f(y),$ then $f$ is a constant function.

Fundamental Theorem of Calculus . Why is this wrong

show that the recursive sequence converges to $\sqrt r$

A question regarding connected subsets of connected metric spaces

Let$ f : [0,1] \to [0,1] $ be a strictly increasing function. Show that it has a fixed point.

Given $\sum\limits_{k=1}^\infty \gamma_k r_k<\infty$ and $\gamma_k\not\in\ell^1$, then $\lim\inf r_k=0$

Is this sequence converging monotonically?

Which of the following of subspaces of $\mathbb{R^2}$ are compact?

A real function sending bounded sets to bounded sets is necessarily of bounded variation?

Continuity of composition of two-variables and one-variable functions

Orthogonality and norm of Hermite polynomials

Trending Questions

Popular # Hahtags

Popular Questions