  An appropriate description of the state of matter that appears as a second
order phase transition is tuned toward zero temperature, {\it viz.}
quantum-critical point (QCP), poses fundamental and still not fully answered
questions. Experiments are needed both to test basic conclusions and to guide
further refinement of theoretical models. Here, charge and entropy transport
properties as well as AC specific heat of the heavy-fermion compound
CeRh$_{0.58}$Ir$_{0.42}$In$_5$, measured as a function of pressure, reveal two
qualitatively different QCPs in a {\it single} material driven by a {\it
single} non-symmetry-breaking tuning parameter. A discontinuous sign-change
jump in thermopower suggests an unconventional QCP at $p_{c1}$ accompanied by
an abrupt Fermi-surface reconstruction that is followed by a conventional
spin-density-wave critical point at $p_{c2}$ across which the Fermi surface
evolves smoothly to a heavy Fermi-liquid state. These experiments are
consistent with some theoretical predictions, including the sequence of
critical points and the temperature dependence of the thermopower in their
vicinity.
