Extending improper Riemann integrability to the endpoint zero

Let \(f:[0,1]\to\mathbb R\) be Riemann integrable on \([b,1]\) for every \(0<b\le1\).

1. If \(f\) is bounded, prove that \(f\) is Riemann integrable on \([0,1]\).
2. What can happen if \(f\) is unbounded?