Fundamental Theorem of Calculus | Expansion of the Theorem [dark version]

TL;DR

Lebesgue integration allows the fundamental theorem to be discussed under weaker pointwise behavior while preserving agreement with Riemann integrals when they exist.

Briefing Cornell Notes

Briefing

The fundamental theorem of calculus can be extended far beyond continuously differentiable functions—but only up to a precise boundary. For a function F on an interval [a,b], the identity

∫[a to b] F′(x) dx = F(b) − F(a)

holds not just when F′ is continuous everywhere, but for a larger class where derivatives exist only almost everywhere and where F′ may fail to behave nicely at some points. The key takeaway is that continuity plus “differentiable almost everywhere” still isn’t enough: there are continuous, monotone functions (built from the Cantor function) whose derivative is zero almost everywhere yet whose total increase is nonzero, breaking the theorem.

The discussion starts with the standard real-analysis version: when F is continuously differentiable on a compact interval, F′ is continuous, the integral of F′ is well-defined, and the theorem links differentiation and integration exactly. It then motivates using the Lebesgue integral (LEC integral) rather than the Riemann integral, because Lebesgue integration is more flexible: whenever a function is Riemann integrable, it matches its Lebesgue integral, but Lebesgue theory also supports weaker assumptions about integrability and pointwise behavior.

From there, the assumptions are weakened in stages. First, the derivative need not exist at every point: since altering a function at finitely many points doesn’t affect the integral, it’s enough that F′ exists almost everywhere—meaning the set of points where F′ fails has Lebesgue measure zero (finite and countable sets qualify). However, even this is insufficient. A simple jump function illustrates the failure: F is constant on [0,1) and jumps at x=1, so F′ exists almost everywhere and equals 0 where defined, making ∫ F′ = 0, while F(1) − F(0) = 1. The theorem fails.

Next comes the more subtle counterexample: the Cantor function. It is continuous and nondecreasing, yet it increases from 0 to 1 while having derivative 0 almost everywhere. That combination—continuity plus differentiability almost everywhere—still doesn’t force the fundamental theorem to hold.

To recover the theorem, the transcript moves to a stronger regularity condition on the derivative: piecewise continuous differentiability. By splitting [a,b] into finitely many subintervals and requiring F to be continuously differentiable on each piece, the proof reduces to applying the standard theorem on each subinterval. The resulting expressions telescope, leaving only F(b) − F(a).

Finally, the maximal extension is framed as an equivalence: for a given F, the fundamental theorem holds on every subinterval [a,c] (for all c in the interval) exactly when F is absolutely continuous. Absolutely continuous functions sit between merely continuous functions and continuously differentiable ones: they are regular enough to guarantee that integrating F′ reconstructs F via F(c) − F(a), and they are precisely the functions for which the generalized fundamental theorem works.

Cornell Notes

The fundamental theorem of calculus links integration of a derivative to the net change of the original function. Standard proofs assume F is continuously differentiable, but the assumptions can be weakened using Lebesgue integration and “almost everywhere” differentiability. Still, continuity plus differentiability almost everywhere is not enough: the jump-function example and the Cantor function show cases where F′=0 almost everywhere (or is undefined at a key point) while F changes anyway, so ∫F′ does not match F(b)−F(a). The theorem does hold under piecewise continuous differentiability, where the interval can be split into finitely many pieces and the standard theorem is applied on each piece with a telescoping sum. The maximal characterization is that the identity holds on every subinterval exactly when F is absolutely continuous.

Why does switching from the Riemann integral to the Lebesgue integral matter for weakening the fundamental theorem’s assumptions?

Lebesgue integration is more general: whenever a function is Riemann integrable, its Riemann and Lebesgue integrals agree. That means one can keep the same conclusions for classical integrable functions while also allowing integrals to be defined under weaker pointwise behavior. This flexibility is crucial when derivatives fail to exist on sets of measure zero, because Lebesgue integration effectively ignores those exceptional points.

What does “derivative exists almost everywhere” mean, and why is it a natural first weakening?

“Almost everywhere” means the set of points where the derivative fails has Lebesgue measure zero. The transcript notes that finite and even countable sets have measure zero. Since integrals are insensitive to changes on measure-zero sets, requiring F′ only almost everywhere is the first step toward weakening the classical “everywhere differentiable” assumption.

How does the jump-function example show that “almost everywhere differentiable” still doesn’t guarantee the theorem?

Consider a piecewise constant function that equals 0 up to x=1 and equals 1 after the jump. Its derivative exists everywhere except at the jump point x=1, and where it exists it is 0. The integral of F′ over the interval therefore gives 0, but the net change F(1)−F(0) equals 1. The mismatch demonstrates that ignoring a single point where differentiability fails can still break the fundamental theorem.

Why does the Cantor function defeat the idea that continuity plus differentiability almost everywhere should be enough?

The Cantor function is continuous and monotonically increasing, yet it increases from 0 to 1 while its derivative equals 0 almost everywhere. So even though F′ exists almost everywhere and is integrable with value 0 almost everywhere, integrating F′ yields 0, while F(b)−F(a) is nonzero. This shows that continuity and almost-everywhere differentiability alone cannot force the integral identity to hold.

What regularity condition restores the fundamental theorem, and how does the proof work?

Piecewise continuous differentiability restores it. The interval [a,b] is split into finitely many subintervals so that on each piece F is continuously differentiable. The standard fundamental theorem applies on each subinterval, producing terms of the form F(A_{j+1})−F(A_j). Summing these gives a telescoping series where intermediate values cancel, leaving only F(b)−F(a).

What is the maximal extension of the theorem, stated as an equivalence?

The transcript gives a characterization: the fundamental theorem holds for F exactly when F is absolutely continuous. Equivalently, for every c in the interval, the identity ∫[a to c] F′(x) dx = F(c) − F(a) must hold (assuming the integral of F′ exists). Absolutely continuous functions are precisely those for which this works on every subinterval.

Review Questions

Give two counterexamples showing why continuity plus differentiability almost everywhere is insufficient for the fundamental theorem. What property fails in each?
Explain how piecewise continuous differentiability leads to a telescoping sum in the proof.
State the maximal condition (in terms of function regularity) under which the generalized fundamental theorem holds on every subinterval.

Key Points

1
Lebesgue integration allows the fundamental theorem to be discussed under weaker pointwise behavior while preserving agreement with Riemann integrals when they exist.
2
“Derivative exists almost everywhere” means the set of nondifferentiability points has Lebesgue measure zero, so integrals ignore those exceptions.
3
A jump discontinuity can break the theorem even when the derivative exists almost everywhere, because the function’s net change can occur at the exceptional point.
4
The Cantor function shows that continuity plus differentiability almost everywhere still fails: it increases while having derivative 0 almost everywhere.
5
Piecewise continuous differentiability restores the theorem by splitting the interval into finitely many pieces and applying the classical result on each piece.
6
The generalized fundamental theorem holds on every subinterval exactly for absolutely continuous functions.

Highlights

A derivative that exists almost everywhere can still produce the wrong integral identity if the function’s change concentrates at exceptional points.

The Cantor function is continuous and nondecreasing, yet its derivative is 0 almost everywhere—so ∫F′ cannot recover F(b)−F(a) without stronger regularity.

Splitting [a,b] into finitely many intervals where F is continuously differentiable makes the proof work via telescoping cancellations.

Absolutely continuous functions are exactly the class for which the fundamental theorem holds in its maximal form.

Topics

Fundamental Theorem of Calculus
Lebesgue Integration
Almost Everywhere Differentiability
Cantor Function
Absolutely Continuous Functions