The following multiple choice question was asked in my exam but I don't know how to proceed:
Define $f:[0,1]\to [0,1]$ by $\displaystyle f(x)=\frac{2^{k}-1}{2^{k}}$ for $\displaystyle x\in [\frac{2^{k-1}-1}{2^{k-1}},\frac{2^{k}-1}{2^{k}}],k\geq 1$. Then $f$ is a Riemann-integrable function such that
1.$\displaystyle \int_{0}^{1}f(x)dx=\frac{2}{3}$
2.$\displaystyle \frac{1}{2}<\int_{0}^{1}f(x)dx<\frac{2}{3}$
3.$\displaystyle \int_{0}^{1}f(x) dx=1$
4.$\displaystyle \frac{2}{3}<\int_{0}^{1}f(x)dx<1$. I know that function is said to be Riemann-integrable if its upper Riemann-integral and lower Riemann-integral exists and are same.