= Solution
{c}
The call spread in the hint has payoff
$$
H_\epsilon(s)=\frac{(s-K)^+-(s-K-\epsilon)^+}{\epsilon}.
$$
It is between zero and one and converges to $1_{\{s>K\}}$ as $\epsilon\downarrow0$. In the <Black-Scholes model> with positive volatility and $t<T$, the conditional terminal stock distribution has no atom at $K$. The strict and non-strict indicator therefore have the same discounted expectation. The <dominated convergence theorem> in the risk-neutral pricing formula gives
$$
V_{\rm digital}
=\lim_{\epsilon\downarrow0}
\frac{C(t,S_t;K)-C(t,S_t;K+\epsilon)}{\epsilon}
=\boxed{-C_K(t,S_t;K).}
$$
Equivalently this is $e^{-r(T-t)}\Phi(d_2)$. The no-atom condition explains the endpoint subtlety: the displayed spread has value zero at $s=K$, whereas the printed digital pays one there. At maturity the strike derivative can be one-sided or undefined, so the smooth pricing identity is understood before expiry.
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