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

Email: knill@math.harvard.edu

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/(xBut note that the function f _{x} is not continuous everywhere. It is the function
g(x,y) = (-(x
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