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A step function integrated by additivity over subintervals, and the same value from the definition

Example

Let $f : [0, 2] \to R$ be

$f (x) : = {\begin{cases} 1 & 0 \leq x < 1, \\ 3 & 1 \leq x \leq 2. \end{cases}$

Then $f$ is integrable on $[0, 2]$ and

$\int_{0}^{2} f = 1 + 3 = 4.$

Two routes are carried out below. The first splits at the jump with For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ and evaluates each piece with Changing an integrable function at finitely many points changes neither its integrability nor its integral and the constant case of If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ . The second computes lower and upper Darboux sums directly on a three-point partition and squeezes.

Why this is the smallest interesting example. The jump sits at the interior point $1$ , which is a partition point of both routes, and a partition point lies in two subintervals (Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions). That is exactly the case singled out in the proof of Changing an integrable function at finitely many points changes neither its integrability nor its integral, and it is why the value of $f$ at the single point $1$ is irrelevant to both the integrability and the value while the upper sums still see it.

Facts & Assumptions

Given: The function $f$ above on $[0, 2]$ , and a real $η$ with $0 < η < 1$ .

[L1]

A monotone function on a closed bounded interval with distinct endpoints is bounded and integrable there (A monotone function on $[a, b]$ is Riemann integrable: for the uniform partition into $N$ parts the upper minus lower sum telescopes to $∣ f (b) - f (a) ∣ (b - a) / ι (N)$ , Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $R$ , with the dictionary to monotone sequences, The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ).

[L2]

For $0 < c < 2$ : $f$ is integrable on $[0, 2]$ if and only if it is integrable on $[0, c]$ and on $[c, 2]$ , and then $\int_{0}^{2} f = \int_{0}^{c} f + \int_{c}^{2} f$ (For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ , The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ).

[L3]

Changing an integrable function at finitely many points changes neither its integrability nor its integral (Changing an integrable function at finitely many points changes neither its integrability nor its integral).

[L4]

$\int_{p}^{q} c = c (q - p)$ for a constant $c$ , and the five-term chain $L (u, P) \leq \int_{‾} u \leq \int^{‾} u \leq U (u, P)$ holds for every partition (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ , The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ).

[L5]

For a partition $P = (n, t)$ and bounded $u$ : $L (u, P) = \sum_{i < n} m_{i} Δ_{i}$ and $U (u, P) = \sum_{i < n} M_{i} Δ_{i}$ with $m_{i} = \inf u [I_{i}]$ , $M_{i} = \sup u [I_{i}]$ , and a set with a least element has it as its infimum (For bounded $f$ on $[a, b]$ and a partition $P$ : the infimum $m_{i}$ and supremum $M_{i}$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L (f, P) = \sum_{i} m_{i} Δ_{i}$ and $U (f, P) = \sum_{i} M_{i} Δ_{i}$ , Finite sums and finite products, by recursion, Laws of finite sums and finite products, Greatest lower bound (infimum), Maximum and minimum of a set, Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions).

[L6]

Ordered-field arithmetic: the order is total and transitive, and a real that is $\leq 4 + 2 η$ for every $η$ in $(0, 1)$ is $\leq 4$ (Ordered field, Complete ordered field (least-upper-bound property), Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Verification

technique · direct

1.1

$f$ is nondecreasing on $[0, 2]$ : for $x \leq y$ the only possibility not giving $f (x) = f (y)$ is $x < 1 \leq y$ , where $f (x) = 1 < 3 = f (y)$ . Hence $f$ is bounded and integrable on $[0, 2]$ , and likewise on $[0, 1]$ and on $[1, 2]$ , by [L1].

givenL1

1.2

Second route. Let $P_{η}$ be the partition of $[0, 2]$ with point set ${0, 1 - η, 1, 2}$ , so $n = 3$ with $Δ_{0} = 1 - η$ , $Δ_{1} = η$ and $Δ_{2} = 1$ .

givenL5construct

2.1

First route, the piece $[0, 1]$ . The constant function $1$ on $[0, 1]$ agrees with $f$ at every point of $[0, 1]$ except $x = 1$ , a single point; so by [L3] and [L4], $\int_{0}^{1} f = \int_{0}^{1} 1 = 1 \cdot (1 - 0) = 1$ .

step 1.1L3L4

2.2

The piece $[1, 2]$ . There $f$ is the constant $3$ , so $\int_{1}^{2} f = 3 \cdot (2 - 1) = 3$ by [L4].

step 1.1L4

2.3

On $I_{0} = [0, 1 - η]$ , $f$ is constantly $1$ , so $m_{0} = M_{0} = 1$ . On $I_{1} = [1 - η, 1]$ , $f$ takes the value $1$ at every point below $1$ and $3$ at $1$ , so $m_{1} = 1$ and $M_{1} = 3$ by [L5]. On $I_{2} = [1, 2]$ , $f$ is constantly $3$ , so $m_{2} = M_{2} = 3$ .

step 1.2givenL5

3.1

By [L2] with $c : = 1$ , $\int_{0}^{2} f = \int_{0}^{1} f + \int_{1}^{2} f = 1 + 3 = 4$ .

step 1.1step 2.1step 2.2L2

3.2

Hence $L (f, P_{η}) = 1 (1 - η) + 1 \cdot η + 3 \cdot 1 = 4$ and $U (f, P_{η}) = 1 (1 - η) + 3 η + 3 \cdot 1 = 4 + 2 η$ , by [L5].

step 2.3L5

4.1

By [L4] the number $\int_{0}^{2} f$ lies between $L (f, P_{η}) = 4$ and $U (f, P_{η}) = 4 + 2 η$ for every $η \in (0, 1)$ , so $4 \leq \int_{0}^{2} f \leq 4$ by [L6]; the second route agrees with step 3.1.

step 1.1step 3.1step 3.2L4L6∎

Remarks

The two sums of $P_{η}$ are not symmetric, and that is a statement about these partitions. For each $η \in (0, 1)$ the lower sum $L (f, P_{η})$ is exactly $4$ while the upper sum is $4 + 2 η > 4$ , so this family already locates the integral exactly from below while its upper sums only approach it. Nothing is claimed here about an arbitrary partition: a partition whose points do not include $1$ has a subinterval on which $f$ takes both values, and its lower sum is then strictly below $4$ .
The value at the jump is irrelevant to the integral, but not to the sums. Replacing $f (1) = 3$ by any real leaves $f$ integrable with the same integral, by Changing an integrable function at finitely many points changes neither its integrability nor its integral. What it changes is which partitions realise the extremes: with $f (1) : = 1$ the subinterval $[1 - η, 1]$ carries $m_{1} = M_{1} = 1$ , while $[1, 2]$ now carries $m_{2} = 1$ and $M_{2} = 3$ , so $L (f, P_{η}) = 2$ and $U (f, P_{η}) = 4$ and the asymmetry moves to the other side. The integral is $4$ either way.
Additivity is what makes the first route short. Without For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ the only route is the second one, which needs the Darboux sums of a function that is not constant on any partition containing the jump in the interior of a subinterval.

Depends on

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Sources

Step function (Wikipedia) (standard reference, not scraped)
Riemann integral (Wikipedia) (standard reference, not scraped)