Theorem 4. Every Riemann integrable function is bounded.

Can we have a bounded function which is not Riemann integrable?

However, once we recall that Riemann integrable functions must be bounded, an example of a derivative that is not Riemann integrable is close at hand. For example, the derivative of the function F defined by F(x) = x2 sin(1/x2) for x = 0 and F(0) = 0 exists at all points, but the function F is not bounded on [0, 1].

Are square integrable functions bounded?

Yes, an integrable function can be unbounded. For example, the function 1/√x on the domain (0,1] is unbounded but the integral has a finite value.