A theorem by Zygmunt Zahorski states that a necessary and sufficient condition for a subset of $\mathbb{R}$ to be the nondifferentiability set of a **continuous** real function is that it is the union of a $G_\delta$ set and a $G_{\delta \sigma}$ set of zero measure.

On the other hand it is not hard to see that the nondifferentiability set of an arbitrary real function is always a $G_{\delta \sigma}$ set.

My question is: why can't we leave the word 'continuous' out in Zahorski's theorem? In other words, what would be an example of a (necessarily discontinuous) real function whose nondifferentiability set is not a union of a $G_\delta$ set and a $G_{\delta \sigma}$ set of zero measure?