I'm studying this theorem on Zorich II:
(Convergence of the Fourier integral at a point) Let $f : \mathbb{R} → \mathbb{C}$ be an absolutely integrable function that is piecewise continuous on each finite closed interval of the real axis $\mathbb{R} $. If the function f satisfies the Dini conditions at a point $x ∈ R$, then its Fourier integral $$\int_{-\infty}^\infty \widehat{f}(\xi) e^{i\xi x}\mathrm{d}\xi$$ converges at that point to the value $(f (x^−) + f (x^+))/2$, equal to half the sum of the left and right-hand limits of the function at that point.
I can't understand why the hypothesis of the piecewise continuity is necessary. It is used only to say that, for every finite $A>0$ and $B>0$:
$$\int_{-A}^A \left(\int_{-B}^B f(t)e^{-i\xi t} \mathrm{d}t\right) e^{i\xi x \mathrm{d}\xi}=\int_{-B}^B f(t) \left(\int_{-A}^A e^{i\xi (x-t)} \mathrm{d}\xi\right)\mathrm{d}t$$
Can't I say the same thing using Fubini theorem?