= Solution
For an integer $m$ and $d\in\{-1,0,1\}$, verify the single-step identity
$$
(m+d)^+-m^+=f(m)d+\frac12\mathbf1_{\{m=0\}}d^2.
$$
If $m\geq1$, the <positive part> is linear across the step and $f(m)=1$; if $m\leq-1$, it remains zero and $f(m)=0$. At $m=0$, $d^+=(d+d^2)/2$, which is checked for the three permitted values of $d$.
Apply this with $m=S_{t-1}-K$ and $d=S_t-S_{t-1}$, and telescope over $t$. The result is the <discrete Tanaka formula>
$$
\boxed{(S_T-K)^+=(S_0-K)^++\sum_{t=1}^Tf(S_{t-1}-K)\Delta S_t+\frac12\sum_{t=1}^T\mathbf1_{\{S_{t-1}=K\}}(\Delta S_t)^2.}
$$
The first sum is a <martingale transform>; the second records the correction at the kink of the <positive part>. The algebraic identity itself needs only the integer values and permitted increments, not the <martingale> property.
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