When modelling strong gravitational lenses, i.e., where there are multiple
images of the same source, the most widely used parameterisation for the mass
profile in the lens galaxy is the singular power-law model $\rho(r)\propto
r^{-\gamma}$. This model may be insufficiently flexible for very accurate work,
for example measuring the Hubble constant based on time delays between multiple
images. Here we derive the lensing properties - deflection angle, shear, and
magnification - of a more adaptable model where the projected mass surface
density is parameterised as a continuous two-dimensional broken power-law
(2DBPL). This elliptical 2DBPL model is characterised by power-law slopes
$t_1$, $t_2$ either side of the break radius $\theta_\mathrm{B}$. The key to
the 2DBPL model is the derivation of the lensing properties of the truncated
power law (TPL) model, where the surface density is a power law out to the
truncation radius $\theta_\mathrm{T}$ and zero beyond. This TPL model is also
useful by itself. We create mock observations of lensing by a TPL profile where
the images form outside the truncation radius, so there is no mass in the
annulus covered by the images. We then show that the slope of the profile
interior to the images may be accurately recovered for lenses of moderate
ellipticity. This demonstrates that the widely-held notion that lensing
measures the slope of the mass profile in the annulus of the images, and is
insensitive to the mass distribution at radii interior to the images, is
incorrect.