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 infinite-dimensional. It is called the Frechet derivative
. The derivative is basically a linear function (usually called "operator" in infinite-dimensional spaces) which approximates the original function locally. Surprisingly, linear functions need not be continuous in the infinite-dimensional 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 2006-08-29 12:13:02