  We use a carefully selected subsample of 1053 confirmed exoplanets from the
NASA Exoplanet Archive to construct empirical power-law exoplanet
mass-radius-temperature ($M$-$R$-$T$) relations. Using orthogonal distance
regression to account for errors in both mass and radius, we allow the data to
decide: 1) the number of distinct planetary regimes; 2) whether the boundaries
of these regimes are best described by broken power laws joined at mass break
points, or by discontinuous power laws motivated by changes in equations of
state and temperature. We find strong support from the data for three distinct
planetary $M$-$R$ regimes and for those regimes to be discontinuous. Our most
successful model involves an $M$-$R$-$T$ relation in which ice/rock (rocky) and
ice-giant (neptunian) planets are segregated by a pure-ice equation of state,
whilst neptunes and gas giant (jovian) planets are segregated by a mass break
at $M_{\rm br} = 115\pm19~M_{\oplus}$. The rocky planet regime is shown to
follow $M \propto R^{0.34\pm0.01}$, whilst neptunes have $M\propto
R^{0.55\pm0.02}$. Planets in both regimes are seen to extend to similar maximum
masses. In the jovian regime, we find that $M \propto R^{0.00\pm0.01}T^{0.35\pm
0.02}$, where $T$ is the planet equilibrium temperature. This implies that, for
jovian planets detected so far, equilibrium temperature alone provides a robust
estimator of mass.
