Let c ∈[a, b] and f:[a, b] →ℝ be given by f(x):={ 0 if a ≤ x ≤ c 1 if c<x ≤

step 1 Okay, so we know that the function F of X is integrable. And if F of X is lessen or equal to zer

Let c ∈[a, b] and f:[a, b] →ℝ be given by f(x):={ 0 ...

Let $c \in[a, b]$ and $f:[a, b] \rightarrow \mathbb{R}$ be given by $$ f(x):=\left\{\begin{array}{ll} 0 & \text { if } a \leq x \leq c \\ 1 & \text { if } c<x \leq b \end{array}\right. $$ Show from first principles that $f$ is integrable on $[a, b] .$ Also, prove that this follows from Proposition $6.10$.

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

Key Concepts

Discontinuities and Measure Zero

In the context of Riemann integration, a function can be discontinuous and still integrable provided the set of points where it is discontinuous has measure zero. This is often formalized in a proposition like Proposition 6.10, which states that if the discontinuities of a bounded function form a set of measure zero, the function is Riemann integrable. In the given problem, the function has a jump discontinuity at a single point, which is a set of measure zero, ensuring the function’s integrability.

Piecewise Constant (Step) Functions

A step function is one that takes on a constant value on subintervals of its domain. This simpler structure makes calculating upper and lower sums more manageable. Since the function only 'jumps' at a finite number of points, one can often show that the effect of these jumps on the integrability is negligible when partitions are chosen carefully.

Riemann Integrability

Riemann integrability is the concept that a bounded function on a closed interval is integrable if the upper and lower Riemann sums can be made arbitrarily close by choosing an appropriate partition. In other words, for every ? > 0, there exists a partition of the interval such that the difference between the upper sum and the lower sum is less than ?.

Upper and Lower Riemann Sums

These sums are constructed by taking the supremum and infimum of the function on each subinterval of a partition. The upper Riemann sum uses the supremum (greatest value) of the function in each subinterval, while the lower sum uses the infimum (least value). The ability to control the difference between these sums is central to demonstrating Riemann integrability from first principles.

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