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Differentiating the European call option price in strike gives, for $T>0$,The strike derivative uses dominated convergence; the second uses the continuous density. Also,Differentiate the supplied time-integral identity and the discount factor. Continuity of the density supplies the diffusion-term derivative. For the tail first moment, continuity in time follows from continuous stock paths, locally uniformly bounded second moments, and the absence of an atom at $K$. ThereforeHence the Dupire equation isIts initial condition is $C(0,K)=(S_0-K{)}^{+}$; natural strike boundaries are $C(T,0)=S_0$ and $C(T,K)\to 0$ as $K\to \infty$. These are consistent with the discounted stock martingale and integrable tails. Where $C_{KK}>0$, the same identity gives local volatility recovery from call prices:The equation evolves in maturity and strike, unlike the backward option-value equation in calendar time and spot.