Partial derivatives do not imply continuity
= Partial derivatives do not imply continuity
Define $f(x,y)=xy/(x^2+y^2)$ away from the origin and zero at the origin. Both <partial derivatives> exist everywhere and are zero at the origin, yet along $x=y$ the function is $1/2$, so it is not <continuous> there. Coordinate-axis derivative tests do not control all neighboring directions.