Sheng Li (gmachine1729) wrote,
Sheng Li

On the fundamental theorem of calculus

Originally published at 狗和留美者不得入内. You can comment here or there.

In [1], we defined the Riemann integral on intervals of [公式] . We shall now prove some theorems pertaining to it. Below, we will let [公式] denote the Riemann sum of [公式] associated with tagged partition [公式].

Definition 1 A piecewise function [公式] defined on a bounded interval [公式] is defined by partitioning [公式] into a finite number of sub-intervals [公式] and defining on each of the [公式] s a function [公式] such that for [公式] , [公式] . We say that [公式] is piecewise continuous if each of the [公式] s is continuous. One can, here, replace “continuous” with any other qualification of function, such as “smooth”.

Proposition 1 We have the following properties of Riemann integrable functions.

  1. If [公式] is Riemann integrable on two disjoint intervals [公式] , with the Riemann integrals equal to [公式] , then [公式] is Riemann integrable on [公式] , with[公式]
  2. Let [公式] be an interval containing interval [公式] . If [公式] is Riemann integrable on [公式] , it is Riemann integrable on both [公式] and [公式] , with [公式]
  3. If on any interval [公式] , both [公式] and [公式] are Riemann integrable, then [公式] is also Riemann integrable on [公式] , with [公式] . Moreover for any [公式] , [公式] is Riemann integrable on [公式] with [公式] . In other word, the Riemann integral on [公式] operator is a linear operator on the space of Riemann integrable functions on [公式] .

Proof: For (1), we take a tagged partition [公式] of [公式] satisfying [公式] and a tagged partition [公式] of [公式] satisfying [公式] . Formally, this says that any [公式] refinement of [公式] is such that [公式] and similar for [公式] . For [公式] we use the tagged partition [公式] . One easily verifies with the triangle inequality that any refinement [公式] of it satisfies [公式] .

For (2), take any tagged partition [公式] of [公式] that contains the endpoints of [公式] . The equivalence of (3) of Lemma 3 from [2] with Riemann integrability tells us that for any [公式] , there exists a refinement [公式] of [公式] such that for any [公式] which are refinements of [公式] ,

[公式] Both of [公式] can be split into tagged partitions of [公式] and [公式] , which we denote via [公式] and [公式] (where the second index represents whether or not the partition is defined on [公式] or [公式] , such that

[公式] If by contradiction [公式] were not Riemann integrable on one of [公式] , then there would by Lemma 3 of [2] exist an [公式] for which there does not exist a partition [公式] that is a refinement of [公式] such that the split of any two refinements of it into partitions on [公式] , which we denote as [公式] and [公式] , satisfies both

[公式] Setting [公式] and using the triangle inequality on the above would thus violate the existence of a refinement [公式] of [公式] which guarantees [公式] , which would imply that [公式] is not Riemann integrable on [公式] , a contradiction.

With similar logic, one proves Riemann integrability of [公式] on [公式] . The integral equality of [公式] then follows directly from [公式] .

The proof of (3) we leave as an exercise to the reader. [公式]

Proposition 2 Any continuous function on [公式] is Riemann integrable.

Proof: We construct a sequence of tagged partitions of [公式] . We let the [公式] th tagged partition be given by


in which we have let [公式] be the subintervals of [公式] corresponding to the partition, where [公式] . As for the tags, we let [公式] , noting that by the Extreme Value Theorem, the infimum here exists. We notice that under the partial order of partition refinement, this sequence of tagged partitions is a chain. Moreover, the Riemann sums of these partitions form a bounded (by Extreme Value Theorem) monotonically non-decreasing sequence. Thus, by the Monotone Convergence Theorem, this sequence of Riemann sums converges to some [公式] . For any [公式] , we can find a partition from this sequence such that the distance between its Riemann sum and [公式] is less than [公式] . Since a refinement of a tagged partition where the tags correspond to infimum of [公式] on subintervals cannot decrease the Riemann sum, [公式] satisfies the definition of Riemann integral. We emphasize in this proof that the subintervals in the definition of Riemann sum are closed and are not disjoint at endpoints, which allows for two tags (of different subintervals) to have the same value. [公式]

Corollary 1 Any function [公式] continuous and bounded on an open or half-open interval in [公式] is Riemann integrable. Moreover, if we extend such a function to the closure of that interval via the limit and Riemann integrate this on the closure, the result is the same. Formally, [公式] . Because of this, we can disregard whether or not the endpoints are included in the interval itself and simply use [公式] to denote this Riemann integral.

Proof: We note that [公式] and [公式] are both closed intervals on which the Riemann integral exists and is zero. The Riemann integral exists on [公式] by Proposition 2. Applying (2) of Proposition 1 tells us that the Riemann integral exists and is equal to [公式] on each of [公式] . [公式]

Definition 2 Suppose that [公式] . Then, for any Riemann integrable [公式] on [公式] we define [公式]

Proposition 3 Any function [公式] piecewise continuous on any interval is Riemann integrable on that interval.

Proof: Corollary 1 tells that we can WLOG only consider closed intervals. Let the interval be [公式] , with [公式] , where the [公式] s are disjoint intervals on each of which [公式] is continuous. Proposition 2 tells us that [公式] is Riemann integrable on each of the [公式] s. Thus, we can apply (1) of Proposition 1 to derive that [公式] is Riemann integrable on [公式] .

One can also use the equivalence of (2) and (3) in Theorem 1 of [2], of which this is a very special case. [公式]

Proposition 4 On some interval [公式] , define Riemann integrable [公式] such that[公式] for all [公式] . Then,


Proof: Trivial and left to the reader.

Theorem 1 (Mean value theorem for definite integrals) Let [公式] be a continuous function. Then, there exists [公式] such that [公式] Proof: The Extreme Value Theorem gives us infimum and supremum values of [公式] on [公式] , which we denote with [公式] and [公式] . This gives us

[公式] By the Intermediate Value Theorem, for every point in [公式] is such that there exists [公式] such that [公式] . Applying this to the value of [公式] yields the desired result. [公式]

Theorem 2 (Fundamental theorem of calculus, part I) Let [公式] be a continuous real-valued function defined on closed interval [公式] . Let [公式] be the function defined, for all [公式] , by

[公式] Then [公式] is uniformly continuous on [公式] and differentiable on the open interval [公式] , and

[公式] for all [公式] .

Proof: Take arbitrary [公式] . We can define for all [公式] such that [公式]

[公式] Applying Mean Value Theorem for Definite Integrals (Theorem 1) to this gives some [公式] such that


As [公式] , [公式] . Because [公式] is continuous on [公式] , we thus have


By the Extreme Value Theorem, there exists [公式] such that [公式] for all [公式] . Thus, by Proposition 4, for any [公式] , take [公式] and we have [公式] for all [公式] . This shows uniform continuity. [公式]

Definition 3 We say that [公式] is an antiderivative of [公式] on [公式] if on [公式] , [公式] .

Lemma 1 Any continuous antiderivative [公式] of the zero function on [公式] is a constant function on [公式].

Proof: Suppose by contradiction that some [公式] that is a continuous anti-derivative of the zero function on [公式] is not constant. Then there exists [公式] such that [公式] . The mean value theorem for differentiable functions tells us that there exists some [公式] such that [公式] , which means that [公式] would then not be a continuous function on [公式] with derivative equal to zero on [公式] . [公式]

Lemma 2 If a function [公式] defined on [公式] has antiderivatives [公式] on [公式] , then [公式] is a constant function on [公式] , where the values of [公式] at [公式] are simply defined via continuous extension.

Proof: The Fundamental Theorem of Calculus, Part I (Theorem 1) tells us that

[公式] is an antiderivative of [公式] on [公式] that is continuous on [公式] . Because the relation defined via constant difference is transitive, it suffices to simply show that [公式] is constant on [公式] .

That [公式] on [公式] tells us that [公式] is an antiderivative of the zero function on [公式] , from which we deduce via Lemma 1 that is a constant function. [公式]

Corollary 2 If [公式] is a real valued continuous function on [公式] and [公式] is a continuous antiderivative of [公式] on [公式] , then


Proof: We define [公式] on [公式] . Fundamental Theorem of Calculus Part I tells us that it is a continuous antiderivative of [公式] on [公式] . We have that


By Lemma 2, [公式] is a constant function on [公式] , we must have that [公式] , which completes our proof. [公式]

Theorem 2 (Fundamental theorem of calculus part II (Newton-Leibniz axiom)) If [公式] is a real valued function on [公式] and Riemann integrable on [公式] and [公式] is an antiderivative of [公式] on [公式] then


Proof: Take an arbitrary partition [公式] . We have that

[公式] We can apply the mean value theorem for differentiability on the subintervals [公式] to assign tags such that [公式] which yields the Riemann sum


By definition of Riemann integrable as given in (4) of Theorem 1 of [2], for all [公式] , there exists [公式] such that for any tagged partition [公式] of [公式] denoted by [公式] and [公式] for the tags such that its norm or mesh [公式] , we have


Another way to put it, per Lemma 2, is that for all [公式] , there exists [公式] such that any tagged partitions [公式] with meshes both less than [公式] ,


such that


Thus, we can, setting [公式] , take the limit of [公式] to derive

[公式] which completes our proof. [公式]


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