I want to prove that \begin{align} f:[0&,1]\to \Bbb{R}\\&x\mapsto \begin{cases}0,&0\leq x<1/2,\\\\ 1,&1/2\leq x\leq 1. \end{cases}\end{align} is Riemann integrable over $[0,1]$ and also, I want to find the value.
MY TRIAL
Let $\epsilon>0,\;x_1\in [0,1/2)$ and $x_2\in [1/2,1]$ such that $x_2-x_1<\epsilon.$ The set $$P=\{0,x_1,x_2,1 \}$$ forms a partition of $[0,1].$ Define $\Delta x_j=(x_j-x_{j-1})$ and $I_j=[x_{j-1}-x_j],\;j=1,2,3.$ So, the upper and lower Darboux sums are given by $$U(f,P)=\sum^{3}_{j=1}M_j \Delta x_j=1-x_1,\;\;\text{where}\;\;M_j=\sup_{x\in I_j}f(x),$$ $$L(f,P)=\sum^{3}_{j=1}m_j \Delta x_j=1-x_2.$$ Then,$$U(f,P)-L(f,P)=x_2-x_1<\epsilon.$$ Hence, $f$ is Riemann integrable over $[0,1]$.
I have been able to show, as seen above, that $f$ is Riemann integrable over $[0,1]$ but not able to find the value of the integral. I know that the answer is $1/2$ but can anyone help out by showing how I could arrive at it?
For each $n\in\mathbb N$, let $P_n=\left\{0,\frac12,\frac12+\frac1{2n},1\right\}$. Compute $U(f,P_n)$ and $L(f,P_n)$ for each natural $n$. What do you get?