  A functional distance ${\mathbb H}$, based on the Hausdorff metric between
the function hypographs, is proposed for the space ${\mathcal E}$ of
non-negative real upper semicontinuous functions on a compact interval. The
main goal of the paper is to show that the space $({\mathcal E},{\mathbb H})$
is particularly suitable in some statistical problems with functional data
which involve functions with very wiggly graphs and narrow, sharp peaks. A
typical example is given by spectrograms, either obtained by magnetic resonance
or by mass spectrometry. On the theoretical side, we show that $({\mathcal
E},{\mathbb H})$ is a complete, separable locally compact space and that the
${\mathbb H}$-convergence of a sequence of functions implies the convergence of
the respective maximum values of these functions. The probabilistic and
statistical implications of these results are discussed in particular,
regarding the consistency of $k$-NN classifiers for supervised classification
problems with functional data in ${\mathbb H}$. On the practical side, we
provide the results of a small simulation study and check also the performance
of our method in two real data problems of supervised classification involving
mass spectra.
