0
votes
0answers
26 views

Question about a theorem concerning the continuity of integral functions

If we have $$F(t):=\int_V f(t,x)dx$$ where $V$ is some measurable subset of $\mathbb R$ and $x\mapsto f(t,x)$ is a measurable function. Moreover let $F$ be defined for all $t\in U$ a open subset of ...
1
vote
2answers
77 views

Show the function is integrable and find the integral - somewhat complex question

We are given $Q = [0,1]$x$[0,1]$ We are also given the function $f(x,y) = (\frac{1}{10})^n$ where $\frac{1}{2^{n+1}} < \max(x,y) \leq \frac{1}{2^n}, (n=0,1,2,...)$ and $f(0,0)=0$. Show that $f$ ...
0
votes
2answers
43 views

Prove that $f^y$ and $f_x$ are Lebesgue-integrable

Let $f:\Bbb R^2\to \Bbb R$ given by: $$f(x,y) = \begin{cases} \frac{x^2-y^2}{(x^2+y^2)^2} & \text{if $(x,y)\in(0,1)\times(0,1)$} \\ 0 & \text{if $(x,y)\not\in(0,1)\times(0,1)$} \\ ...
1
vote
1answer
122 views

Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
1
vote
1answer
45 views

Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$ g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
0
votes
1answer
73 views

prove $\exists$ a continuous function g s.t. $|g(x)-f(x)|<\epsilon$ for all x in a subset E of (a,b) and $\mu[(a,b)-E]<\delta$

I am having trouble starting with ths problem. Let f be a Lebesgue-integrable function over a bounded interval (a,b). Prove that for any $\epsilon >0$, $\delta >0$, there exists a continuous ...
2
votes
1answer
330 views

Generalization of absolute continuity with $f(x) = x^a \sin(1/x^b)$

As a generalization of Prove that $x^\alpha \cdot\sin(1/x)$ is absolutely continuous on $(0,1)$ : Let $f : (0, 1] \to \mathbb{R}$ be the function denoted by $f(x) = x^a \sin(1/x^b)$. Determine for ...
2
votes
2answers
147 views

Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
1
vote
1answer
81 views

Continuity and Integrability

Any help with this problem is appreciated. Consider the function $f(x) = \sum_{n=1}^\infty x n^{-\beta} e^{-nx}$. For what values of $\beta \in \mathbb{R}$ is $f$ continuous on $[0,\infty)$ and in ...
3
votes
1answer
4k views

{$\int_{[1/n,1]}f$} to converge and yet $f$ is not $L$-integrable over $[0,1]$

Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$. I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$. My attempts: I've found ...
0
votes
1answer
142 views

What does Luzin's theorem imply?

Luzin's theorem states that: let $f:[a,b]\rightarrow R$ be an a.e. finite function, $f$ is measurable iff $\forall \epsilon \geq 0: \exists \phi_\epsilon$ continuous on $[a,b]$ and $\mu\{x: f(x)\neq ...