  It is shown how piecewise differentiable functions $F: \mathbb R^n \mapsto
\mathbb R^m $ that are defined by evaluation programs can be approximated
locally by a piecewise linear model based on a pair of sample points $\check x$
and $\hat x$. We show that the discrepancy between function and model at any
point $x$ is of the bilinear order $O(\|x-\check x\| \|x-\hat x\|)$. This is a
little surprising since $x \in \mathbb R^n$ may vary over the whole Euclidean
space, and we utilize only two function samples $\check F=F(\check x)$ and
$\hat F=F(\hat x)$, as well as the intermediates computed during their
evaluation. As an application of the piecewise linearization procedure we
devise a generalized Newton's method based on successive piecewise
linearization and prove for it sufficient conditions for convergence and
convergence rates equaling those of semismooth Newton. We conclude with the
derivation of formulas for the numerically stable implementation of the
aforedeveloped piecewise linearization methods.
