For a bounded function $f:[a,b] \rightarrow \mathbb{R}$, we define the lower Riemann-integral of $f$ over $[a,b]$ as $\underline{S}(f)= \sup \{ \underline{S}(f,P) | P$ is a partition of $[a,b] \}$. The upper Riemann-integral is defined as $\overline{S}(f)= \inf \{ \overline{S}(f,P) | P$ is a partition of $[a,b] \}$.
I am asked to give an example for which strictly $\underline{S}(f)< \overline{S}(f)$. Is the following function a good choice?; $f:[a,b] \rightarrow \mathbb{R}: x \mapsto \begin{cases} 1 & \text{for } x \in \mathbb{Q} \cap [a,b],\\ 0 & \text{for } x \in (\mathbb{R} \backslash \mathbb{Q}) \cap [a,b]\\ \end{cases}$