I have recently come across a result, which states that if $F$ is the distribution function of a random variable $X$ with characteristic function $C$, then:
$$\sum\left(\nabla{F(x)}\right)^{2}=\lim_{T\to\infty}\frac{1}{2T}\int^{T}_{-T}\left|C(t)\right|^{2}\,\mathrm{d}t$$
where $\nabla{F(x)}$ the $\mathbb{P}(X=x)$. Well, if I straightforward accept this, then I've proven that the integrability of $\left|C(t)\right|^{2}\Longrightarrow{F}$ is continuous! Can someone give an elementary proof of the result I've stated above (in bold-italics)?