Given $f\in R (\mathbb{T}),$ Prove $\sum_{n=-\infty}^\infty|\hat f(n)|= \infty$

Question

Let $f\in R (\mathbb{T})$ be a continuous function in every $x\neq x_0$ with period $2\pi$.

Suppose that at $x_0,$ the one-sided limits exist and different one from the other.

Prove: $\sum_{n=-\infty}^\infty|\hat f(n)|=\infty$

First, I tried to work with Bessel's inequality, but couldn't really get far.

Second, I tried to work with Fejér's theorem, but again, got stuck.

Any help appreciated.

Answer 1

Hint: Assume the sum is finite and prove that $f$ must be continuous also at $x_0$.