  This note provides a counterexample to a theorem announced in the last part
of the paper ''Analysis of direct searches for discontinuous functions'',
Mathematical Programming Vol. 133, pp.~299--325, 2012. The counterexample
involves an objective function $f: \mathbb{R} \to \mathbb{R}$ which satisfies
all the assumptions required by the theorem but contradicts some of its
conclusions. A corollary of this theorem is also affected by this
counterexample. The main flaw revealed by the counterexample is the possibility
that a directional direct search method (dDSM) generates a sequence of trial
points $(x_k)_k$ converging to a point $x_*$ where $f$ is discontinuous and
whose objective function value $f(x_*)$ is strictly less than
$\lim_{k\to\infty} f(x_k)$. Moreover the dDSM generates no trial point in one
of the two branches of $f$ near $x_*$. This note also investigates the proof of
the theorem to highlight the inexact statements in the original paper. Finally
this work concludes with a modification of the dDSM that allows to recover the
properties broken by the counterexample.
