  In this article we investigate the century-old continuous extension problem
of the Riemann map. Let $G$ be a simply connected domain. We call $\lambda$ in
$\partial G$ a multiple point if there are simply connected subdomains $ U$ and
$V$ such that $\lambda \in\partial U \cap\partial V$ and $ dist (\partial U\cap
G , \partial V\cap G )>0$. We show that the Riemann map of $G$ has a continuous
extension to $\overline G$ if and only if $\partial G$has no multiple points.
All of the results in this paper, together with the Riemann mapping theorem,
give a complete and desirable solution to the mapping problem that was
originally raised by Riemann in 1851 and intensively investigated by many
famous mathematicians throughout history.
