  The Dirac fermions at the surface of a topological insulator can be gapped by
introducing magnetic dopants. Alternatively, in an ultra-thin slab with
thickness on the order of the extent of the surface states, both the top and
bottom surface states acquire a common gap value ($\Delta$) but with opposite
sign. In a topological insulator, the dominant piece of the Hamiltonian
($\hat{H}$) is of a relativistic nature. A subdominant non-relativistic piece
is also present and in an external magnetic field ($B$) applied perpendicular
to the surface, the $N=0$ Landau level is no longer at zero energy but is
shifted to positive energy by the Schr{\"o}dinger magnetic energy. When a gap
is present, it further shifts this level down by $-\Delta$ for positive
$\Delta$ and up by $|\Delta|$ for a negative gap. This has important
consequences for the magneto-optical properties of such systems. In particular,
at charge neutrality, the lowest energy transition displays anomalous
non-monotonic behaviour as a function of $B$ in both its position in energy and
its optical spectral weight. The gap can also have a profound impact on the
spectral weight of the interband lines and on corresponding structures in the
real part of the dynamical Hall conductivity. Conversely, the interband
background in zero field remains unchanged by the non-relativistic term in
$\hat{H}$ (although its onset frequency is modified).
