Darboux Fundamental Theorem of Calculus

by Ng Tze Beng

It was well known that if a function f is differentiable on [a, b] and if f ' is Riemann integrable, then

.

The usual proof of this fact is to use the Mean Value Theorem. The proof is very easy following the Theorem in Riemann Integral and Sum of Infinite Series and using a special Riemann sum with respect to a regular partition and a choice of the points satisfying the Mean Value Theorem in each interval.

We shall give a proof without Mean Value Theorem.

We shall restate Theorem 2 and Theorem 2' in Do we need Mean Value Theorem to prove f ' (x) = 0 on (a, b) implies that f = constant on (a, b)? as follows.

Theorem 1. If f :[a, b] ® R is differentiable, then for any u, v in [a, b] with u < v, there exists a point x and a point y in [u, v] such that

or equivalently, .

Proof of Darboux Theorem.

We are going to make use of Theorem 1 to define two sequences of Riemann sums that both converge to the Riemann integral .

For each integer n > 1 define the regular partition P_n

P_n : a = x₀ < x₁< ... < x_n = b ,

with || P_n || = (b-a)/n, for k = 0, 1, ¼, n. For each k = 1, ¼, n by Theorem 1, there exist x _k and h_k in [x_k-1, x_k] such that

f ' (x _k ) (x_k - x_k-1) £ f (x_k) - f (x_k-1) £ f ' (h _k ) (x_k - x_k-1).

Define S_n to be the Riemann sum with respect to the partition P_n witth the choice of points in each subinterval [x_k-1, x_k] given by x _k . Then

• .

Similarly we define T_n to be the Riemann sum with respect to the partition P_n witth the choice of points in each subinterval [x_k-1, x_k] given by h _k . Then we also have

• . Thus

by the Theorem in Riemann Integral and Sum of Infinite Series,

• .

Similarly, we have .

Therefore, .

ã Ng Tze Beng 2001