Example of an integrable, but discontinuous function
Define by . For g we have:
g is differentiable and .
is discontinuous (at 0).
1. ► If we see that g is differentiable at x due to the product and the chain rule [7.6.3/11]. The derivation at x calculates to
If we need to check if the difference quotient function has a limit. But as the following estimate holds for all (note that sin is bounded by 1)
the identity is valid, so that g is differentiable at 0 with .
2. ► As e.g. , but , we find to be discontinuous at 0.
So we know that is discontinuous and has a primitive.