  Serre proved that any holomorphic cusp form of weight one for $\Gamma_1(N)$
is lacunary while a holomorphic modular form for $\Gamma_1(N)$ of higher
integer weight is lacunary if and only if it is a linear combination of cusp
forms of CM-type (see Serre, subsections 7.6 and 7.7). In this paper, we show
that when a non-zero modular function of arbitrary real weight for any finite
index subgroup of the modular group ${\SL}_2(\Z)$ is lacunary, it is
necessarily holomorphic on the upper-half plane, finite at the cusps and has
non-negative weight.
