Existence of partial derivatives does not imply their continuity
= Existence of partial derivatives does not imply their continuity
A function can be differentiable even when its partial derivatives are discontinuous. For example,
$$
f(x,y)=(x^2+y^2)\sin\frac1{\sqrt{x^2+y^2}}
$$
away from the origin, with value zero at the origin, is differentiable there while its partial derivatives oscillate along the coordinate axes.