Let $h : [0,1] \to \mathbb{R}$ a measurable function such that $h$ is not equal almost everywhere to a continuous function.
Question. Show that there exists a $\lambda >0$ such that for every integer $n \ge 1$, $x \mapsto \exp(2i \pi n \lambda h(x))$ is not equal almost everywhere to a continuous function.
I have barely no ideas. How can I do that ? Thanks.
Not a full solution, but perhaps will be useful for ideas. Define $h = \chi_{[1/2,1]}.$ This is a basic example on $[0,1]$ that is not equal to any continuous function a.e. Now we look at $\exp (2\pi i h(x)).$ That function is identically $1,$ so $\lambda = 1$ will not work. And in fact no rational $\lambda$ will work. But suppose $\lambda$ is irrational. Then $\exp (2\pi in\lambda h(x))$ does the job for every $n.$ That's because $2\pi i n\lambda h(x)=0$ on $[0,1/2),$ and equals $2\pi i n\lambda$ on $[1/2,1].$ Since $n\lambda$ is never an integer, exponentiating will give a function equal to $1$ on $[0,1/2),$ and a constant not equal to $1$ on $[1/2,1].$