This page uses the standard upper-semicontinuous subharmonic convention

The subharmonic functions on this page take values in $[-\mathrm{\infty },\mathrm{\infty })$, are required to be upper semicontinuous, and are excluded from being identically $-\mathrm{\infty }$ on any connected component of the domain. This is the standard potential-theoretic convention used by the sources behind the page, and it is the one compatible with $\log \mathrm{\mid }f\mathrm{\mid }$ for a holomorphic function and with Perron's method for the Dirichlet problem.

The older convention that a subharmonic function is merely a continuous real-valued function satisfying the submean inequality is recovered as the special case where the function never takes the value $-\mathrm{\infty }$ and happens to be continuous. The harmonic comparison theorems on the page are written so that the harmonic notion from Plane harmonic functions remains the same while the subharmonic class is large enough to include logarithmic singularities.