Prove $ F(x)=\int_0^{\infty}\frac{f(y)}{x+y}dy $ is continuous on $(0,\infty)$ and differentiable, and have $\lim\limits_{x\to \infty} F(x)=0$. $f$ is integrable on $[0,\infty)$, and $\int_0^{\infty} |f(y)|dy < \infty$.
Prove:
Then $ F(x)=\int_0^{\infty}\frac{f(y)}{x+y}dy $ is continuous on $(0,\infty)$ and differentiable, and have $\lim\limits_{x\to \infty} F(x)=0$.

A brief proof


*

*the function $x\mapsto \frac{f(y)}{x+y}$ is continuous on $(0,+\infty)$

*$\lim_{x\to\infty}\frac{f(y)}{x+y}=0$

*$\forall a>0$ and if $x\geq a$ we have $\frac{|f(y)|}{x+y}\leq\frac{|f(y)|}{a+y}\leq\frac{1}{a}|f(y)|\quad(1)$


then use dominated convergence theorem and you have an answer for 2 points.


*

*the function $x\mapsto\frac{f(y)}{x+y}$ is differentiable and $x\mapsto-\frac{f(y)}{(x+y)^2}$ its derivative then use the same idea as in $(1)$ and the dominated convergence theorem and you have an answer for the last point

