6760, Math 21a, Fall 2009
Exhibits page Math 21a 2009, Multivariable Calculus
Discontinuous function for which partial derivatives exist
Course head: Oliver Knill
Office: SciCtr 434

Discontinuous function for which partial derivatives exist

The function f

The partial derivative fx

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 fx 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.
Questions and comments to knill@math.harvard.edu
Math21b | Math 21a | Fall 2009 | Department of Mathematics | Faculty of Art and Sciences | Harvard University