Can you find a function that is differentiable at the origin but the function itself is not continuous at the origin?
Here's another stab at it: Discontinuous functions can be differentiated when considering "generalized functions" called distributions
. When we use this concept, we can e.g. differentiate the Heaviside step function
which is discontinuous at the origin and obtain the dirac delta
as the derivative. Since we can (sort of) evaluate the dirac delta at the origin (getting infinity as function value), one could say the Heaviside step function can be differentiated at the origin.
Well allright, sounds more like metaphysics than math, maybe this is stretching the terms continuity and differentiability a little too far...
Posted by JLo
on 2006-09-25 18:58:16