  Aims. The aim of the study is to constrain the physics of gamma-ray bursts
(GRBs) by analysing the multi-wavelength afterglow data set of GRB 121024A that
covers the full range from radio to X-rays. Methods. Using multi-epoch
broad-band observations of the GRB 121024A afterglow, we measured the three
characteristic break frequencies of the synchrotron spectrum. We used six
epochs of combined XRT and GROND data to constrain the temporal slopes, the
dust extinction, the X-ray absorption, and the spectral slope with high
accuracy. Two more epochs of combined data from XRT, GROND, APEX, CARMA, and
EVLA were used to set constraints on the break frequencies and therefore on the
micro-physical and dynamical parameters. Results. The XRT and GROND light
curves show a simultaneous and achromatic break at around 49 ks. As a result,
the crossing of the synchrotron cooling break is no suitable explanation for
the break in the light curve. The multi wavelength data allow us to test two
plausible scenarios explaining the break: a jet break, and the end of energy
injection. The jet-break scenario requires a hard electron spectrum, a very low
cooling break frequency, and a non-spreading jet. The energy injection avoids
these problems, but requires $\epsilon_e > 1 (k = 2)$, spherical outflow, and
$\epsilon_B < 10^{-9}$. Conclusions. In light of the extreme microphysical
parameters required by the energy-injection model, we favour a jet-break
scenario where $\nu_m < \nu_{sa}$ to explain the observations. This scenario
gives physically meaningful microphysical parameters, and it also naturally
explains the reported detection of linear and circular polarisation.
