Piecewise Continuous Function with One-Sided Limits is Darboux Integrable/Proof 2

Proof
We are given that $f$ is piecewise continuous with one-sided limits on $\left[{a \,.\,.\, b}\right]$.

Therefore, there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that for all $i \in \left\{{1, 2, \ldots, n}\right\}$:


 * $f$ is continuous on $\left({x_{i - 1} \,.\,.\, x_i}\right)$
 * $\displaystyle \lim_{x \mathop \to {x_{i - 1} }^+} f \left({x}\right)$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} f \left({x}\right)$ exist.

Note that $n$ is the number of intervals $\left({x_{i - 1} \,.\,.\, x_i}\right)$ defined from the (finite) subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$.

We shall use proof by induction on the intervals.

For all $k \in \left\{{1, 2, \ldots, n}\right\}$, let $P \left({k}\right)$ be the proposition:
 * $f$ is Riemann integrable on $\left[{x_0 \,.\,.\, x_k}\right]$.

Basis for the Induction
$P \left({1}\right)$ is the case:
 * $f$ is Riemann integrable on $\left[{x_{i - 1} \,.\,.\, x_i}\right]$

for an arbitrary $i \in \left\{{1, 2, \ldots, k}\right\}$.

Piecewise continuity with one-sided limits of $f$ for the case $n = 1$ means that:
 * $f$ is continuous on $\left({x_{i - 1} \,.\,.\, x_i}\right)$
 * $\displaystyle \lim_{x \mathop \to {x_{i - 1} }^+} f \left({x}\right)$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} f \left({x}\right)$ exist.

By Integrability Theorem for Functions Continuous on Open Intervals, $f$ is Riemann integrable on $\left[{x_{i - 1} \,.\,.\, x_i}\right]$.

Thus $P \left({1}\right)$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is the induction hypothesis:
 * $f$ is Riemann integrable on $\left({x_0 \,.\,.\, x_k}\right)$.

from which it is to be shown that:
 * $f$ is Riemann integrable on $\left({x_0 \,.\,.\, x_{k + 1} }\right)$.

Induction Step
This is the induction step:

By definition of a piecewise continuous function with one-sided limits, for every $i \in \left\{ {1, 2, \ldots, k, k + 1}\right\}$:


 * $f$ is continuous on $\left({x_{i - 1} \,.\,.\, x_i}\right)$


 * the one-sided limits $\displaystyle \lim_{x \mathop \to x_{i - 1}^+} f \left({x}\right)$ and $\displaystyle \lim_{x \mathop \to x_i^-} f \left({x}\right)$ exist.

By the induction hypothesis, $f$ is Riemann integrable on $\left[{x_0 \,.\,.\, x_k}\right]$.

From the basis for the induction, $f$ is Riemann integrable on $\left[{x_k \,.\,.\, x_{k + 1} }\right]$.

We have that $f$ is Riemann integrable on $\left[{x_0 \,.\,.\, x_k}\right]$ and $\left[{x_k \,.\,.\, x_{k + 1} }\right]$.

Therefore, by Existence of Integral on Union of Adjacent Intervals:
 * $f$ is Riemann integrable on $\left[{x_0 \,.\,.\, x_k}\right] \cup \left[{x_k \,.\,.\, x_{k + 1} }\right]$.

We have that:
 * $\left[{x_0 \,.\,.\, x_{k + 1} }\right] = \left[{x_0 \,.\,.\, x_k}\right] \cup \left[{x_k \,.\,.\, x_{k + 1} }\right]$

Accordingly, $f$ is Riemann integrable on $\left[{x_0 \,.\,.\, x_{k + 1} }\right]$.

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $f$ is Riemann integrable on $\left[{a \,.\,.\, b}\right]$