  We discuss dynamical response functions near quantum critical points,
allowing for both a finite temperature and detuning by a relevant operator.
When the quantum critical point is described by a conformal field theory (CFT),
conformal perturbation theory and the operator product expansion can be used to
fix the first few leading terms at high frequencies. Knowledge of the high
frequency response allows us then to derive non-perturbative sum rules. We
show, via explicit computations, how holography recovers the general results of
CFT, and the associated sum rules, for any holographic field theory with a
conformal UV completion -- regardless of any possible new ordering and/or
scaling physics in the IR. We numerically obtain holographic response functions
at all frequencies, allowing us to probe the breakdown of the asymptotic
high-frequency regime. Finally, we show that high frequency response functions
in holographic Lifshitz theories are quite similar to their conformal
counterparts, even though they are not strongly constrained by symmetry.
