(Piecewise constant functions). Let [a, b] be an interval. A...

(Piecewise constant functions). Let $[a, b]$ be an interval. A piecewise constant function $f:[a, b] \rightarrow \mathbf{R}$ is a function for which there exists a partition of $[a, b]$ into finitely many intervals $I_1, \ldots, I_n$, such that $f$ is equal to a constant $c_i$ on each of the intervals $I_i$. If $f$ is piecewise constant, show that the expression $$ \sum_{i=1}^n c_i\left|I_i\right| $$ is independent of the choice of partition used to demonstrate the piecewise constant nature of $f$. We will denote this quantity by p.c. $\int_a^b f(x) d x$, and refer to it as the piecewise constant integral of $f$ on $[a, b]$.

(Piecewise constant functions). Let $[a, b]$ be an interval. A piecewise constant function $f:[a, b] \rightarrow \mathbf{R}$ is a function for which there exists a partition of $[a, b]$ into finitely many intervals $I_1, \ldots, I_n$, such that $f$ is equal to a constant $c_i$ on each of the intervals $I_i$. If $f$ is piecewise constant, show that the expression

$$
\sum_{i=1}^n c_i\left|I_i\right|
$$

is independent of the choice of partition used to demonstrate the piecewise constant nature of $f$. We will denote this quantity by p.c. $\int_a^b f(x) d x$, and refer to it as the piecewise constant integral of $f$ on $[a, b]$.

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