I read the following from Chung's Probability:
($f$ is the characteristic function of some distribution function $F$) Suppose that $f''(0)$ exists and is finite, then we have $f''(0)=\lim\limits_{h \to 0}\frac{f(h)-2f(0)+f(-h)}{h^{2}}$
Could anyone help on how to show this? The problem is that we only assume $f$ is twice differentiable at the point 0. So we can't prove the above simply by Taylor's expansion.
Thank you very much!