## Show $f$ is infinitely differentiable February 7

Assume that $E$ is a compact Lebesgue measurable subset of $\mathbb{R}$ and let, $$f(t) = \int_E \cos(tx) \, dx, \hspace{1mm} t \in \mathbb{R}$$Show that $f$ is infinitely differentiable.I have a solution using the mean value theorem, induction, and