  The exact exchange-correlation (xc) kernel $f_\mathrm{xc}(x,x',\omega)$ of
linear response time-dependent density functional theory is computed over a
wide range of frequencies, for three canonical one-dimensional finite systems.
Methods used to ensure the numerical robustness of $f_\mathrm{xc}$ are set out.
The frequency dependence of $f_\text{xc}$ is found to be due largely to its
analytic structure, i.e. its singularities at certain frequencies, which are
required in order to capture particular transitions, including those of double
excitation character. However, within the frequency range of the first few
interacting excitations, $f_\mathrm{xc}$ is approximately frequency
independent, meaning the exact adiabatic approximation $f_\mathrm{xc}(\omega =
0)$ remedies the failings of the local density approximation and random phase
approximation for these lowest transitions. The key differences between the
exact $f_\mathrm{xc}$ and its common approximations are analyzed, and cannot be
eliminated by exploiting the limited gauge freedom in $f_\mathrm{xc}$. The
optical spectrum benefits from using as accurate as possible an $f_\mathrm{xc}$
and ground-state xc potential, while maintaining exact compatibility between
the two is of less importance.
