Showing that a function is Darboux Integrable using definition of integral

Question

I have been given a function $f:[0, 4]→ \mathbb{R}$ such that $f(x) = 0$ for all $x \ne 2$ and $f(2) = 2$, and told to show that $f$ is Darboux integrable in $[0,4]$. However, I don't understand how a function like this can be integrable, as it is mostly a constant function that equals $0$ and then a single disconnected point at $(2, 2)$. How is this function meant to be integrable?

Answer 1

Let $\{\mathcal P_n\}$ be a sequence of partitions of $[0,4]$ such that $\operatorname{gap}\mathcal P_n \rightarrow 0$

It is trivial to show that $\mathcal L(f,\mathcal P_n) = 0$ for any $n$. Then, you just need to show that $\mathcal U(f,\mathcal P_n) \rightarrow 0$, which you can do by using the fact that $\mathcal U(f,\mathcal P_n) \le 2 \cdot \operatorname{gap}\mathcal P_n$.