Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 3 / 3B / ii

On the nonzero $x$ axis the function equals one, whereas its assigned value at the origin is zero. Therefore $f(x,0)$ does not tend to $f(0,0)$ as $x\to 0$. It is not a continuous function there, and differentiability implies continuity shows