$\Vert f \Vert_p$  is continuous in $p$? When $f$ is a Lebesgue measurable function, is $\Vert f \Vert_p$ continuous in $p$, when $p$ is in a set such that f belongs to $L_p$? If yes, how to show that? Thanks!
Yes. It is an exercise taken from Rudin's book. I have spent quite a while thinking but got no idea.

This may not be the answer you are looking for, but:
Using Holder's inequality you can show that the map $p^{-1}\mapsto \|f\|_p$ is log-convex. Now just use the fact that any convex function on an interval is continuous. 
