(related to Definition: Riemann-Integrable Functions)
The proceeding propositions show that there are quite a lot of functions which are Riemann integrable. A question arises if there are any functions that are not. The following function $f:[0,1]\to\mathbb R$ is an example of a non-Riemann-itegrable function.
$$f(x):=\begin{cases}1&\text{if }x\text{ is rational}\\0&\text{if }x\text{ is irrational}\end{cases}.$$
The function is bounded on the closed interval $[0,1].$ Howerver, the Riemann upper and lower integrals are not equal each other:
$$\int_{0~*}^{1}f(x)dx=0\quad\neq \quad \int_{0}^{1~*}f(x)dx=1.$$