  We show that the variance of centred linear statistics of eigenvalues of GUE
matrices remains bounded for large $n$ for some classes of test functions less
regular than Lipschitz functions. This observation is suggested by the limiting
form of the variance (which has previously been computed explicitly), but it
does not seem to appear in the literature. We combine this fact with comparison
techniques following Tao-Vu and Erd\"os, Yau, et al. and a Littlewood-Paley
type decomposition to extend the central limit theorem for linear eigenvalue
statistics to functions in the H\"older class $C^{1/2+\epsilon}$ in the case of
matrices of Gaussian convolution type. We also give a variance bound which
implies the CLT for test functions in the Sobolev space $H^{1+\epsilon}$ and
$C^{1-\epsilon}$ for general Wigner matrices satisfying moment conditions.
Previous results on the CLT impose the existence and continuity of at least one
classical derivative.
