= Solution
Since the initial value and increments are integers, $m=S_T-K$ is an integer. If $m\leq-1$, all three <positive parts> in the second difference are zero. If $m\geq1$, the second difference is $(m-1)-2m+(m+1)=0$, including $m=1$. At $m=0$ it equals $0-0+1=1$. Thus
$$
\boxed{(S_T-K-1)^+-2(S_T-K)^++(S_T-K+1)^+=\mathbf1_{\{S_T=K\}}.}
$$
The original PDF has the first term $(S_T-K-1)^+$; the TeX's $(S_T-K+K)^+$ is an extraction error. This <discrete call-price curvature> identity isolates an individual integer state.
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