  Using a calibration method, we prove that, if $w$ is a function which
satisfies all Euler conditions for the Mumford-Shah functional on a
two-dimensional open set $\Omega$, and the discontinuity set of $w$ is a
segment connecting two boundary points, then for every point $(x_0, y_0)$ of
$\Omega$ there exists a neighbourhood $U$ of $(x_0, y_0)$ such that $w$ is a
minimizer of the Mumford-Shah functional on $U$ with respect to its own
boundary values on $\partial U$.
