To change the function into a continuous function, we set
This function is now differentiable at 0, because:
- (f(x) - f(0)) / (x - 0) = x sin( 1 / x )
Since | x sin( 1 / x ) | < | x | for all x, we see that the limit of the difference quotient for c = 0 equals zero. Hence, f is differentiable at 0, and
- f'(0) = 0
The function is also differentiable everywhere else, since it is the product and composition of differentiable functions everywhere but for x = 0. Therefore, the function is differentiable on the whole real line.
Actually, this function is more interesting than it seems, because it is
- once differentiable
- derivative is not continuous
Thus, it provides an example to show that even if derivatives exist, they do not necessarily have to be continuous. These statements are proved at a later point, but you might want to try it on your own already.