## 6.5. Differentiable Functions

### Examples 6.5.6(c):

The function

*f(x) = x*has a removable discontinuity at^{2}sin(1 / x )*x = 0*. If the function is extended appropriately to be continuous at*x = 0*, is it then differentiable at*x = 0*?
To change the function into a continuous function, we set
*f(0) = 0*
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

- continuous
- 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.