PCHIP slope selection has no zero denominator

Each $h_i=x_{i+1}-x_i$ is positive, so every secant $d_i$ is defined and finite. In the two-node branch both slopes equal the already defined secant, and neither endpoint formula is evaluated. When $n\ge 3$, the endpoint denominators $h_0+h_1$ and $h_{n-2}+h_{n-3}$ exist and are positive.

The interior harmonic-mean branch is entered only when $d_{k-1}{d}_k>0$. Thus both secants are nonzero and have the same sign; the weights $w_1=2h_k+h_{k-1}$ and $w_2=h_k+2h_{k-1}$ are positive. The two terms $w_1\mathrm{/}{d}_{k-1}$ and $w_2\mathrm{/}{d}_k$ therefore have that same nonzero sign, so their sum is nonzero and finite. The endpoint clip and cap branches only compare finite real quantities and introduce no division. Hence every output slope is defined and finite.