The function f(x) = | x | is continuous everywhere. Is it also differentiable everywhere ?
We know that f is continuous. To check for differentiability, we have to employ the basic definition:
- f'(x) =
| x | - | c | ) / (x - c)
- If c > 0 then x > 0 eventually. Then there is no need for the absolute value. The limit become +1.
- If c < 0 then x < 0 eventually. The absolute values are resolved by an additional negative sign. The limit becomes -1.
- If c = 0, then the left and the right-handed limits will be different (-1 and +1). Therefore, the function is not differentiable at 0.
This is an example of a function that shows that differentiability
is a stronger concept than continuity: