Let c ∈(a, b) and f:[a, b] →ℝ be given by f(x):={ (x-c) /(a-c) if a ≤ x ≤ c, (x-c

step 1 All right, so the first step here, since F is going to be remand integral by the Archimedes rema

Let c ∈(a, b) and f:[a, b] →ℝ be given by f(x):={ (x-c)...

Let $c \in(a, b)$ and $f:[a, b] \rightarrow \mathbb{R}$ be given by $$ f(x):=\left\{\begin{array}{ll} (x-c) /(a-c) & \text { if } a \leq x \leq c, \\ (x-c) /(b-c) & \text { if } c<x \leq b \end{array}\right. $$ Show from first principles that $f$ is integrable on $[a, b] .$ Also, prove that this follows from Proposition $6.10$. (Hint: For $n \in \mathbb{N}$, consider the partition $P_{n}:=\{a, a+(c-a) / n, \ldots, a+(c-a)(n-1) / n, c, c+(b-c) / n, \ldots$ $c+(b-c)(n-1) / n, b\} .)$

Let $c \in(a, b)$ and $f:[a, b] \rightarrow \mathbb{R}$ be given by
$$
f(x):=\left\{\begin{array}{ll}
(x-c) /(a-c) & \text { if } a \leq x \leq c, \\
(x-c) /(b-c) & \text { if } c<x \leq b
\end{array}\right.
$$
Show from first principles that $f$ is integrable on $[a, b] .$ Also, prove that this follows from Proposition $6.10$. (Hint: For $n \in \mathbb{N}$, consider the partition $P_{n}:=\{a, a+(c-a) / n, \ldots, a+(c-a)(n-1) / n, c, c+(b-c) / n, \ldots$
$c+(b-c)(n-1) / n, b\} .)$

Key Concepts

Piecewise Continuous Functions

Piecewise continuous functions are those which are continuous on each of a finite number of subintervals, even if there may be discontinuities at the endpoints of these subintervals. Such functions are often Riemann integrable because the behavior on the individual continuous pieces allows for controlled estimates of the function’s upper and lower sums and the discontinuities are typically isolated or removable in the integration process.

Upper and Lower Riemann Sums

Upper and lower Riemann sums are used to approach the definition of the integral from above and below. The lower sum uses the infimum of the function on each subinterval while the upper sum uses the supremum. When the difference between these sums can be made arbitrarily small, it is a strong indication of the function’s integrability, since both converge to the same limit, the definite integral.

Sufficient Conditions for Integrability (Proposition 6.10)

Proposition 6.10 typically provides sufficient conditions under which a function is integrable, such as continuity or having a finite number of discontinuities. This proposition reinforces the idea that if a function behaves well—either being continuous or piecewise continuous—then the rigorous definitions of Riemann integration will ensure that the upper and lower sums converge, thereby confirming the function’s integrability.

Riemann Integrability

Riemann integrability is the concept of defining the integral of a function as the limit of Riemann sums over partitions of the interval, provided that this limit exists and is independent of how the partitions are chosen provided they become sufficiently fine. A function is Riemann integrable if the upper and lower sums can be made arbitrarily close by choosing an appropriate partition of the interval.

Partitioning of the Interval

The idea of partitioning an interval involves dividing the domain into a finite collection of subintervals. This approach is used to approximate the area under the curve of the function. By refining the partition—making the subintervals smaller—the sum of the areas of rectangles (or other shapes) constructed over these subintervals converges to the function’s integral.

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