Counterexamples in integral calculus: do functions like these exist?

Question

Could you give me examples of functions of the following kinds?

1. A function which is Riemann-integrable AND has an antiderivative, but is not continuous
2. A function which is Riemann-integrable, and has an antiderivative which does not differ from its integral function up to a constant
3. A function which is not continuous, but such that its integral function is its antiderivative too.