The function f
The partial derivative fx
Continuity can be a bitch (*). You have seen already examples
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 fx is not continuous everywhere. It is the function
g(x,y) = (-(x2y) + y3)/(x2 + y2)2
The above example shows that this statement is not true if one looks only at a single point.