The solution of the Dirac equation for ground-state hydrogen atom (under point nuclei, fixed nuclei position) includes, as quoted in this link, $$ r^{\gamma-1},$$ where $$ \gamma = \sqrt{k^2 - Z^2 \alpha^2 } = \sqrt{ 1 - 1/137^2} < 1,$$ thus it diverges at the origin. Namely not continuous, but still within $L^2$ space.
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