Where is the following function continuous

[firenze]

Homework Statement

f: [0,+\infty) \to \mathbb{R}: y \mapsto \int_0^{+\infty} y \arctan x \exp(-xy)\,dx.<br />
Show that this function is continuous in y if y \neq 0
and discontinuous if y = 0

Homework Equations

The Attempt at a Solution

I just can't get started, any hint?

 

[StatusX]

Start by trying to simplify an expression for f(y+d)-f(y). Ultimately you want to show that for any epsilon>0, you can pick a delta so that |f(y+d)-f(y)|<epsilon for all d<delta.

 

[firenze]

Here is my try:
Choose a sequence y_n \in [0,+\infty ) such that y_n \to y (\neq 0).
Define the function g_n(x)=y_n \arctan x e^{-xy_n}, then its limit is g(x)=y\arctan x e^{-xy}.
Note that |g_n(x)| \leq |y_n\arctan x|, it follows g_n is integrable. Hence by dominated convergence thm we have
\lim f(y_n)=\lim \int g_n \to \int g = f(y).

Am I right? Still no idea for the case y=0

 

[StatusX]

To use the dominated convergence theorem you need to find a function that bounds g_n for all n. In other words, this function can't have an n in it. Other than that you seem to be on the right track. The same idea should work for y=0.