Is there a map $f:[0,1] \mapsto R$ which is unbounded?

Question

The other condition is for any $a \in (0,1), f$ is Riemann integrable on $[a,1]$ with $\int^{1}_{a} f(x) dx =0 $.

Answer 1

Yes. Let $f(\frac{1}{n})=n$ for $n \in \mathbb{N}$ and $f(x)=0$ elsewhere.