Fourier sums for the Banach indicatrix

We prove the existence of a function $f(t)$, which is continuous on the interval $[0,1]$, is of bounded variation, $\min f(t)=0$, $\max f(t)=1$, for which the integral
$$ I(x)=\frac1\pi\int_0^\infty\biggl[\int_0^1\cos y(f(t)-x)|df(t)|\biggr]\,dy $$
diverges for almost all $x\in[0,1]$. This result gives a negative answer to a question posed by Z. Ciesielski.