In the lecture notes (Oxford University, Applied Partial differential equations) it says that $u$ is continuous and only first derivatives of $u$ may be discontinuous across some curve $C$ in the $(x,y)$-plane, while $u$ itself is continuous there. The curve is defined by $(x(t),y(t))$.
Since $u$ is continuous across $C$, so is $\dot{u}$.
I don't understand this? Surely, if $u$ was continuous but had a "corner", which occured along $C$, its derivative wouldn't be continuous? Thanks in advance.