  For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family
$F_\sigma$ $(0<\sigma<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding
maps such that each $F_\sigma$ has an observable (i.e. Lebesgue positive)
Lyapunov irregular set. These maps are obtained by modifying the piecewise
expanding map given in Tsujii (2000). In strong contrast to it, we also show
that any Lyapunov irregular set of any 2-dimensional piecewise real analytic
expanding map is not observable. This is based on the spectral analysis of
piecewise expanding maps in Buzzi (2000).
