The decay of Fourier coefficients and the continuity of functions

Question

Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not know how to use the assumption that $ f $ is of bounded variation. Can you give me some hints?

Answer 1

The (distributional) derivative $\mu=f'$ is a measure, and by assumption $\widehat{\mu_n}=o(1)$. By Wiener's theorem, this implies that $\mu$ does not have a point part, so $\mu$ is a continuous measure and thus $f$ is a continuous function. Compare Corollary 13.11 of my lecture notes here.