Discontinuous function for which partial derivatives exist

The function f

The partial derivative f_x

Continuity can be a bitch (*). You have seen already examples like here, where a function can be discontinuous even so all directional derivatives exist. Hairer and Wanner: Analysis by its history, Springer 2008, page 304. mention that for the following discontinuous discontinuous function all partial derivatives exist at (0,0). This is obvious since for x=0 as well as for y=0 the function is zero.

f(x,y) = xy/(x²+y²)

But note that the function f_x is not continuous everywhere. It is the function

g(x,y)  = (-(x²y) + y³)/(x² + y²)²

If a function f(x,y) has continuous partial derivatives everywhere in the plane, then it is also continuous everywhere in the plane.

The above example shows that this statement is not true if one looks only at a single point.

(*) "Bitch" has been reappropriated to have positive meanings in some contexts.