Can you find a function that is differentiable at the origin but the function itself is not continuous at the origin?
Not quite a solution, but maybe worth a post: Derivatives can also be defined for functions with a domain that is infinitedimensional. It is called the
Frechet derivative. The derivative is basically a linear function (usually called "operator" in infinitedimensional spaces) which approximates the original function locally. Surprisingly, linear functions need not be continuous in the infinitedimensional case. And since the Frechet derivative of a linear function is the function itself, that would mean that such a linear, discontinuous function would be differentiable.
Not quite a solution though, because the strict definition of the Frechet derivative requires that the approximating linear function is bounded, which is equivalent to being continuous, i.e. my fine example would not really count in this strict sense of the definition.

Posted by JLo
on 20060829 12:13:02 