Put-price strike derivative (${\partial }_{K}P(K)=D\mathbb{Q}(S_T<K)$)

For a European put option with deterministic discount factor $D$ and an integrable terminal stock price, $P(K)=D{\mathbb{E}}_{\mathbb{Q}}[(K-S_T{)}^{+}]$. If $\mathbb{Q}(S_T=K)=0$, bounded difference quotients and the dominated convergence theorem give the displayed derivative, the price of a digital put option. At an atom, the left derivative is $D\mathbb{Q}(S_T<K)$ and the right derivative is $D\mathbb{Q}(S_T\le K)$, so the ordinary derivative need not exist.