  We develop a test to determine whether a function lying in a fixed
$L_2$-Sobolev-type ball of smoothness $t$, and generating a noisy signal, is in
fact of a given smoothness $s\geq t$ or not. While it is impossible to
construct a uniformly consistent test for this problem on every function of
smoothness $t$, it becomes possible if we remove a sufficiently large region of
the set of functions of smoothness $t$. The functions that we remove are
functions of smoothness strictly smaller than $s$, but that are very close to
$s$-smooth functions. A lower bound on the size of this region has been proved
to be of order $n^{-t/(2t+1/2)}$, and in this paper, we provide a test that is
consistent after the removal of a region of such a size. Even though the null
hypothesis is composite, the size of the region we remove does not depend on
the complexity of the null hypothesis.
