I'm learning Perron's method and have a question on the definition of subharmonic functions.
Definition Let $\Omega\subset \mathbb{C}$ be a bounded domain. A function u: $\Omega \to [-\infty, +\infty)$ is called subharmonic if it is upper semicontinuous and, for every closed disk $B(a,r) \subset \Omega$, we have the inequality \begin{equation} u(a) \le \frac{1}{2\pi} \int^{2\pi}_0 u(a+re^{i\theta})d\theta. \end{equation}
In different textbooks, for a function $u$ to be subharmonic, someone requires that $u$ is continuous. In the Perron method, I don't see any difference of such two definitions, and each definiton works well to solve the Dirichlet problem.
My question is: what's the difference between these two definitions? I mean, is there any situation that we have to define a subharmonic harmonic to be upper semicontinuous?