Continuous transmission need not preserve normal derivatives ($[u]=0/⟹[{\partial }_{\nu }u]=0$)

A function continuous across an interface can have different one-sided normal derivatives. For instance, $u(x,y)=y$ when $y<0$ and $u(x,y)=0$ when $y\ge 0$ is continuous, satisfies $u_{xx}+\mathrm{sign}(y)u_{yy}=0$ on each open side, and has zero tangential second derivative on the interface. Its lower $y$-derivative is one and its upper $y$-derivative is zero. Thus the normal derivative does not exist there as a two-sided derivative.

More generally, for functions smooth on each side with one-sided first derivative traces, global $C^1$ regularity would force both traces to equal the derivative on the interface. Unequal traces therefore preclude a globally classical $C^2$ solution as well. A condition on the function value alone is not a condition on its normal derivative; a specified transmission law or an appropriate weak formulation is additional information.