Riemann (...Darboux) Integrals

Main ideas and lessons

• The video introduces Riemann/Darboux-style integrals as a rigorous way to define the “area under a curve” by approximating a function using partitions and upper/lower sums.
• A key conceptual distinction (with motivation):
  • Students often confuse integral vs. anti-derivative.
  • This chapter’s goal is to rigorously define integration and later prove results connecting integrals and anti-derivatives.
• The speaker emphasizes that although the course may say “Riemann integral,” the actual definitions/proofs use Darboux integrals, which are equivalent (as stated to be shown in exercises).

---

Step-by-step methodology (definitions and proof tools)

1) Setup: partitions and bounds

• Consider a bounded function (f) on an interval ([a,b]) (including endpoints).

• A partition (P) of ([a,b]) is a finite set of points [ x_0, x_1, \dots, x_n ] such that:

  • (x_0=a), (x_n=b)
  • (x_0 < x_1 < \cdots < x_n)
  • Define subinterval lengths: [ \Delta x_i = x_i - x_{i-1}. ]

---

2) Auxiliary quantities: infimum and supremum on each subinterval

For each subinterval ([x_{i-1}, x_i]):

• Let [ m_i = \inf{f(x) : x \in [x_{i-1}, x_i]} ]

• Let [ M_i = \sup{f(x) : x \in [x_{i-1}, x_i]}. ]

---

3) Darboux sums (lower and upper)

Given partition (P):

• Lower Darboux sum: [ L(P,f)=\sum_{i=1}^n m_i\,\Delta x_i ]

• Upper Darboux sum: [ U(P,f)=\sum_{i=1}^n M_i\,\Delta x_i ]

Geometric interpretation:

• (L(P,f)) is the total area of rectangles with heights given by the lower bounds (m_i).
• (U(P,f)) is the total area of rectangles with heights given by the upper bounds (M_i).
• The partition need not have equal-width subintervals (equal widths are only for drawing convenience).

---

4) Fundamental inequality for any bounded function (Proposition 5.1.2)

Assume:

• (f) is bounded on ([a,b])
• There exist global bounds (m \le f(x) \le M) on ([a,b])

Then for any partition (P):

• Always: [ L(P,f)\le U(P,f). ]

• Also: [ m(b-a)\le L(P,f), \qquad U(P,f)\le M(b-a). ]

Core reasoning described:

• (\sum \Delta x_i = b-a).
• On each subinterval: (m \le m_i \le M_i \le M).
• Combine to get the chain of inequalities.

---

5) Darboux integrals: inf/sup over all partitions (Definition 5.1.3)

Let (\mathcal{P}) be the set of all partitions of ([a,b]).

Define:

• Upper Darboux integral: [ \overline{\int_a^b} f = \inf{U(P,f): P\in\mathcal{P}} ]

• Lower Darboux integral: [ \underline{\int_a^b} f = \sup{L(P,f): P\in\mathcal{P}}. ]

Notes: These always exist for bounded (f), since the lower and upper sums are trapped between (m(b-a)) and (M(b-a)).

---

6) Example showing upper and lower may differ

• Define [ f(x)= \begin{cases} 1, & x \text{ rational}\ 0, & x \text{ irrational} \end{cases} ]

• Claim:

  • (\overline{\int_a^b} f = b-a)
  • (\underline{\int_a^b} f = 0)

Reasoning given:

• Every non-degenerate subinterval contains both rationals and irrationals.
• Therefore, on each subinterval:
  • (\inf f = 0) and (\sup f = 1).
• So for every partition (P):
  • (U(P,f)=1\cdot(b-a))
  • (L(P,f)=0)

Conclusion: Upper and lower Darboux integrals need not match, so not all bounded functions are integrable.

---

7) Refinements: the key convergence tool

Definition (refinement)

• If (P \subseteq \widetilde{P}), then (\widetilde{P}) is a refinement of (P).
• For two partitions (P_1, P_2), their mutual refinement can be taken as (P_1 \cup P_2) (the smallest common refinement).

Important restriction: Refinement only makes sense when the interval ([a,b]) is fixed.

Effect of refining (Proposition 5.1.7)

If (\widetilde{P}) is a refinement of (P), then:

• Lower sums can only move up: [ L(P,f)\le L(\widetilde{P},f) ]

• Upper sums can only move down: [ U(P,f)\ge U(\widetilde{P},f). ]

Proof strategy described:

• Reduce to the case where (\widetilde{P}) is obtained by adding one new point (y) to (P).
• Only one subinterval ([x_{k-1},x_k]) gets split.
• Infimum on a smaller set can only increase; supremum on a smaller set can only decrease.

Consequences (Proposition 5.1.8)

With global bounds (m \le f \le M): [ m(b-a)\le \underline{\int_a^b} f \le \overline{\int_a^b} f \le M(b-a). ]

Also, for any partitions (P_1,P_2), if they share a mutual refinement (\widetilde{P}), then: [ L(P_1,f)\le U(P_2,f). ]

This uses: “All lower sums are (\le) all upper sums,” hence (\sup(\text{lower sums}) \le \inf(\text{upper sums})).

---

8) Darboux/Riemann integrability criterion (final definition)

• A bounded function (f) on ([a,b]) is Darboux integrable (also called “Riemann integrable” per the equivalence mentioned) if: [ \underline{\int_a^b} f = \overline{\int_a^b} f. ]

• The common value is written: [ \int_a^b f(x)\,dx. ]

Immediate bounds: If (m \le f \le M) on ([a,b]), then: [ m(b-a)\le \int_a^b f \le M(b-a). ] A variant noted: if (|f(x)|\le M), then [ \left|\int_a^b f\right|\le M(b-a). ]

---

Examples of integrability

(a) Constant functions

• If (f(x)=c), then lower and upper Darboux integrals coincide.
• Conclusion:
  • Constant functions are integrable
  • [ \int_a^b c\,dx = c(b-a). ]

(b) A sharp jump localized to a point

• Domain: ([0,2])

• Define: [ f(x)= \begin{cases} 0, & x<1\ \tfrac12, & x=1\ 1, & x>1 \end{cases} ]

• Claim: [ \int_0^2 f(x)\,dx = 1. ]

Method described:

• Choose a partition that isolates the “non-constant behavior at/near (x=1)” into a very small middle subinterval of size depending on (\varepsilon).
• For that partition, the difference (U(P,f)-L(P,f)) equals exactly (\varepsilon).
• Since (\varepsilon) is arbitrary, lower and upper integrals coincide.

---

Main integrability test (Proposition 5.1.13) — practical criterion

Statement

Let (f) be bounded on ([a,b]). Then (f) is Darboux integrable iff:

• For every (\varepsilon>0), there exists a partition (P) such that: [ U(P,f)-L(P,f)<\varepsilon. ]

Proof idea (two directions)

If (f) is integrable:

• Since (\underline{\int_a^b} f = \overline{\int_a^b} f), choose partitions where lower sums and upper sums are within (\varepsilon) of that common value.
• Take a mutual refinement so the bounds squeeze tightly.
• Conclude (U(P,f)-L(P,f)) can be made (<\varepsilon).

If such partitions exist for all (\varepsilon):

• Then the supremum of lower sums and infimum of upper sums must coincide because they are trapped inside an interval smaller than (\varepsilon).
• Hence (f) is integrable.

---

Application: integrability of (f(x)=x) on ([a,b])

Goal: show integrable (not compute the integral).

• Pick (\varepsilon>0).

• Choose (n) large enough so that: [ \frac{(b-a)^2}{n}<\varepsilon. ]

• Use a uniform partition: [ x_i = a + i\frac{b-a}{n}, ] so each subinterval has width (\frac{b-a}{n}).

• Since (f(x)=x) is increasing:

  • infimum on each subinterval occurs at the left endpoint,
  • supremum occurs at the right endpoint.

• Compute: [ U(P,f)-L(P,f) = n\left(\frac{b-a}{n}\right)^2 = \frac{(b-a)^2}{n} <\varepsilon. ]

• By Proposition 5.1.13, (f(x)=x) is integrable.

---

Intuition and notation conventions

• Intuition:
  • Integration aggregates local information (values on subintervals) into a global result (area/accumulation over ([a,b])).
  • The integral sign can be viewed as a “stretched s” for summing many local contributions.
  • Integrals reflect global behavior, unlike derivatives (which are highly local).
• Restriction to intervals: If (f) is integrable on a larger domain, its restriction to ([a,b]) is integrable.

• Changing the order of limits:

  • If (a>b), define [ \int_a^b f = -\int_b^a f. ]

  • If (a=b), define [ \int_a^a f = 0. ]

• Dummy variable: The integration variable name does not matter (e.g., (x) vs. (t) vs. (s)).

---

Speakers / sources featured

• No specific named speakers or external sources are mentioned.
• The content is presented by a single instructor (unnamed), describing propositions and definitions from a real analysis chapter.