(Therorem) Continuous functions are Darboux integrable

Theorem:

Let $f:[a,b]\to \mathbb{R}$ be bounded. Then $f$ is Darboux integrable if and only if for all $ϵ>0$, there exists a partition $P$ of $[a,b]$ such that

$$U(f,P)-L(f,P)<ϵ.$$

Proof from Modern Analysis I.

For the forward implication, let $f$ be Darboux integrable. For every $ϵ>0$, there exist partitions $P_1,P_2$ such that

$$U(f,P_1)-\overline{{\int }_a^b}f<ϵ/2$$ $$\underline{{\int }_a^b}f-L(f,P_2)<ϵ/2.$$

Let $P=P_1\cup P_2$ be the common refinement of these partitions, so

$$U(f,P)-L(f,P)\le U(f,P_1)-L(f,P_2)+\underline{{\int }_a^b}f-\overline{{\int }_a^b}f<ϵ.$$

Conversely, suppose that for all $ϵ>0$ there exists a partition $P$ such that the statement of the theorem holds. Then

$$0\le \overline{{\int }_a^b}f-\underline{{\int }_a^b}f\le U(f,P)-L(f,P)<ϵ,$$

so the upper and lower Darboux sums are equivalent. $\square$

Thoerem:

Let $f:[a,b]\to \mathbb{R}$ be continuous. Then $f$ is Darboux integrable.

Proof from Modern Analysis I.

By the previous theorem, it suffices to find a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<ϵ$.

Recall that (Theorem) Continuous functions on compact sets are uniformly continuous, so we can find $\delta >0$ such that for $|x-y|<\delta$, we have

$$|f(x)-f(y)|<\frac{ϵ}{b-a}.$$

Fix a partition $P=\{x_0,\dots ,x_n\}$ of $[a,b]$ such that $|x_k-x_{k-1}<\delta$. By continuity of $f$, for each $k$, we can find some $x_k^{+},x_k^{-}$ such that

$$f(x_k^{+})=\underset{x\in [x_{k-1},x_k]}{\sup }f(x)f(x_k^{-})=\underset{x\in [x_{k-1},x_k]}{\inf }f(x)$$

#wip