Where is the following function continuous

[firenze]

Homework Statement

$$f: [0,+\infty) \to \mathbb{R}: y \mapsto \int_0^{+\infty} y \arctan x \exp(-xy)\,dx.$$
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.