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.

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 (not belong to $C^0$, thus cannot be $C^2$), but still within $L^2$ space.

The solution of Dirac equation for ground-state hydrogen atom includes (under point nuclei  , fixed nuclei position)  https://en.wikipedia.org/wiki/Hydrogen-like_atom#1S_orbital $$ r^{\gamma-1} $$, 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.

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.