  The M dwarf Proxima~Centauri is known to be magnetically active and it hosts
a likely Earth-like planet in its habitable zone. Understanding the
characteristics of stellar radiation by understanding the properties of the
emitting plasma is of paramount importance for a proper assessment of the
conditions on Proxima~Centauri~b and exoplanets around M dwarfs in general. We
determine the temperature structure of the coronal and transition region plasma
of Proxima Centauri from simultaneous X-ray and far-ultraviolet (FUV)
observations. The differential emission measure distribution (DEM) was
constructed for flaring and quiescent periods by analysing optically thin X-ray
and FUV emission lines. Four X-ray observations of Proxima Centauri were
conducted by the LETGS instrument on board of the Chandra X-ray Observatory and
four FUV observations were carried out using the Hubble STIS spectrograph. From
the X-ray light curves, we determined a variation of the quiescent count rate
by a factor of two within 20\% of the stellar rotation period. To obtain the
DEM, 18 optically thin emission lines were analysed (12 X-ray and six FUV). The
flare fluxes differ from the quiescence fluxes by factors of 4-20 (FUV) and
1-30 (X-ray). The temperature structure of the stellar corona and transition
region was determined for both the quiescence and flaring state by fitting the
DEM(T) with Chebyshev polynomials for a temperature range $\log$T = 4.25 - 8.
Compared to quiescence, the emission measure increases during flares for
temperatures below 0.3\,MK (FUV dominated region) and beyond 3.6\,MK (X-ray
dominated region). The reconstructed DEM shape provides acceptable line flux
predictions compared to the measured values. We provide synthetic spectra at
1-1700 \AA, which may be considered as representative for the high-energy
irradiation of Proxima~Cen~b during quiescent and flare periods.
