Extracting the risk-neutral density from call prices

Suppose you observe prices $C(K)$ of European calls with a fixed expiry $T$ for every strike $K$. Show how to recover the risk-neutral probability density of the stock price at $T$.

$f(K)=e^{rT}\,\dfrac{\partial^2C}{\partial K^2}$. Since $C(K)=e^{-rT}\int_K^\infty(s-K)f(s)\,ds$, one derivative in $K$ gives $-e^{-rT}\mathbb{P}(S_T>K)$ and a second gives $e^{-rT}f(K)$ (Breeden-Litzenberger). In practice this is a butterfly spread divided by the squared strike spacing.