Riemann Integrability of Continuous Functions and Functions of Bounded Variation
Recall from the Riemann-Stieltjes Integrability of Continuous Functions with Integrators of Bounded Variation page that we proved that if $f$ is a continuous function on $[a, b]$ and $\alpha$ is a function of bounded variation on $[a, b]$ then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.
We have looked a lot of Riemann-Stieltjes integrals thus far but we should not forget the less general Riemann-Integral which arises when we set $\alpha (x) = x$ since these integrals are fundamentally important in calculus.
The function $\alpha(x) = x$ is a monotonically increasing function and we've already see on the Monotonic Functions as Functions of Bounded Variation page that every monotonic function is of bounded variation.
Consequentially, the following theorem follows rather naturally as a corollary for Riemann integrals from the theorem referenced at the top of this page.
| Theorem 1: Let $f$ be a function defined on $[a, b]$. a) If $f$ is continuous on $[a, b]$ then $\int_a^b f(x) \: dx$ exists. b) If $f$ is of bounded variation on $[a, b]$ then $\int_a^b f(x) \: dx$ exists. |
- Proof of a): Suppose that $f$ is continuous on $[a, b]$. Note that $\alpha(x) = x$ is a function of bounded variation. Hence by the theorem referenced at the top of this page we have that $f$ is Riemann-Stieltjes integrable with respect to $\alpha(x) = x$ on $[a, b]$, that is, $\int_a^b f(x) \: dx$ exists.
- Proof of b): Suppose that $f$ is of bounded variation. Then since $\alpha(x) = x$ is a continuous function, by the theorem above we have that $\alpha$ is Riemann-Stieltjes integrable with respect to $f$ on $[a, b]$. But as we saw on The Formula for Integration by Parts of Riemann-Stieltjes Integrals page, this implies that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$, that is, $\int_a^b f(x) \: dx$ exists. $\blacksquare$
