Does the construction of a continuous function whose Fourier series diverges at one point get any easier if one allows jump discontinuities?

Question

More specifically, I seek a function $f\colon \mathbb{R} \to \mathbb{C}$ that satisfies

1. $f$ is $2\pi$-periodic
2. $f$ is continuous on $[0,2\pi]$ except at finitely many points
3. At every point of $\mathbb{R}$, the left- and right-hand limits of $f$ exist and are finite, and the value of $f$ is their average
4. There exists $x_0 \in \mathbb{R}$ so that the Fourier series of $f$ evaluated at $x_0$ either diverges or converges to a value not equal to $f(x_0)$.

Points 2. and 3. imply that $f$ is determined by its values on any full-measure set.

I know that there are examples (du Bois-Reymond, Fejer, Kolmogorov, etc) of even continuous functions that satisfy 1. and 4. above. I find those examples rather complicated. Does the above weakening of continuity make a simple construction available?