= Solution
Taking <expectations> in the single-step <discrete Tanaka formula>, the <martingale transform> term vanishes because $f$ is bounded and $\mathcal F_T$-measurable. Hence
$$
C(T+1,K)-C(T,K)=\frac12\mathbb E[\mathbf1_{\{S_T=K\}}(S_{T+1}-S_T)^2].
$$
At a state with $p_K=\mathbb P(S_T=K)>0$, the <tower property of conditional expectation> gives $\mathbb E[S_{T+1}\mid S_T=K]=K$, so the conditional second <moment> of the increment is $\sigma^2(T,K)$. Part (a) gives $p_K=C(T,K+1)-2C(T,K)+C(T,K-1)$. Combining these identities yields the <discrete Dupire equation>
$$
\boxed{C(T+1,K)-C(T,K)=\frac12\sigma^2(T,K)\bigl(C(T,K+1)-2C(T,K)+C(T,K-1)\bigr).}
$$
\b[The conditional <variance> needs a positive-probability conditioning state.] The bound $|K-S_0|\leq T$ does not guarantee $p_K>0$. If $p_K=0$, both the <call-price curvature> and the time increment above are zero; the equation can be extended by assigning an arbitrary finite value to $\sigma^2(T,K)$ at such a state, but its conditional <variance> is not determined there.
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