Examples 6.5.6(b):
The function f(x) = x sin(1/x) is continuous
everywhere except at x = 0, where it has a removable
discontinuity. If the function is extended appropriately
to be continuous at x = 0, is it then
differentiable at x = 0 ?
We have seen before that this function has a removable discontinuity
at x = 0. If we set
then the function is continuous on the real line. To check differentiability, we'll have to check the limit of the difference quotient for c = 0. |
(f(x) - f(0) ) / x = x sin( 1 / x) / x = sin( 1 / x )
But we have seen before that this function does not have a limit as x approaches 0. Can you recall the argument for this statement ?
Therefore, the function is not differentiable at 0. As a product and composition of functions that are differentiable for all x but zero, this function is differentiable everywhere except at x = 0.