Is the scalar magnetic potential continuous?

A potential is essentially an integral of work to carry a particle (magnetic or electric one) from an infinite distance up to the point you need to know the potential value. If the force making the work is not a Dirac's Delta, then the integral (i.e. the potential) must be a continuous function.

The potential for a static magnetic problem is defined through $$ {bf B} = - nabla phi $$ (or you can define another potential for $bf H$). Then since $nabla cdot {bf B} = 0$ we have Laplace's equation for $phi$ and that is why it is useful: we have lots of good methods for solving Laplace's equation. (Of course it won't work if $nabla times {bf B} ne 0$; in that case one should adopt another approach.)

The above equation implies that the answer to your question is that $phi$ is continuous if and only if $bf B$ is finite. At a boundary such as a surface you expect to find finite $bf B$. At a current-carrying wire of arbitarily small radius, on the other hand, $bf B$ can diverge.