Can you find a function that is differentiable at the origin but the function itself is not continuous at the origin?

If we read "differentiable" as "partially differentiable", then the following function will do:

f(x,y) = 0 for x=0 and y=0
f(x,y) = xy / ( x^2 + y^2 ) otherwise

The partial derivatives df/dx and df/dy exist everywhere with df/dx=df/dy=0 at the origin. But it isn't continuous because f(x,x)=1/2 for x unequal 0 and f(0,0)=0.

In the mutlivariate case the implication "partially differentiable=>continuous" no longer holds. Thank god we still have "totally differentiable=>continuous".

Posted by JLo on 2006-08-29 10:13:26