Diamagnetism and the dispersion of the magnetic permeability

It is well known that the usual Kramers--Kronig relations for the relative permeability function $\mu(\omega)$ are not compatible with diamagnetism ($\mu(0)<1$) and a positive imaginary part ($\text{Im}\,\mu(\omega)>0$ for $\omega>0$). We demonstrate that a certain physical meaning can be attributed to $\mu$ for all frequencies, and that in the presence of spatial dispersion, $\mu$ does not necessarily tend to 1 for high frequencies $\omega$ and fixed wavenumber $\mathbf k$. Taking the asymptotic behavior into account, diamagnetism can be compatible with Kramers--Kronig relations even if the imaginary part of the permeability is positive. We provide several examples of diamagnetic media and metamaterials for which $\mu(\omega,\mathbf k)\not\to 1$ as $\omega\to\infty$.