Darboux's Theorem

Theorem

Let $f$ be a real function which is bounded on the closed interval $\closedint a b$.

Then:

$\ds m \paren {b - a} \le \underline {\int_a^b} \map f x \rd x \le \overline {\int_a^b} \map f x \rd x \le M \paren {b - a}$

where:

$M$ is an upper bound of $f$ on $\closedint a b$

$m$ is a lower bound of $f$ on $\closedint a b$

$\ds \overline {\int_a^b} \map f x \rd x$ denotes the upper Darboux integral of $f$ on $\closedint a b$

$\ds \underline {\int_a^b} \map f x \rd x$ denotes the lower Darboux integral of $f$ on $\closedint a b$

If $f$ is Darboux integrable on $\closedint a b$, then in fact:

$\ds m \paren {b - a} \le \int_a^b \map f x \rd x \le M \paren {b - a}$

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$.

Suppose that $\forall t \in \closedint a b: \size {\map f t} < \kappa$.

Then:

$\ds \forall \xi, x \in \closedint a b: \size {\int_x^\xi \map f t \rd t} < \kappa \size {x - \xi}$

Consider the finite subdivision of $\closedint a b$ given by:

$P = \set {a, b}$

From their definitions, the upper and lower Darboux sums can be computed as:

$\ds \map {U^{\paren f}} P = \paren {b - a} \sup_{x \in \closedint a b} \map f x$

$\ds \map {L^{\paren f}} P = \paren {b - a} \inf_{x \in \closedint a b} \map f x$

Since by definitions of the supremum and infimum:

$\ds \sup_{x \in \closedint a b} \map f x \le M$

$\ds \inf_{x \in \closedint a b} \map f x \ge m$

we can conclude that:

$\ds \map {U^{\paren f}} P \le \paren {b - a} M$

$\ds \map {L^{\paren f}} P \ge \paren {b - a} m$

Moreover, by definitions of the upper and lower Darboux integrals:

$\ds \overline {\int_a^b} \map f x \rd x \le \map {U^{\paren f}} P$

$\ds \underline {\int_a^b} \map f x \rd x \ge \map {L^{\paren f}} P$

The inequalities stated in the theorem follow by transitivity, along with Upper Darboux Integral Never Smaller than Lower Darboux Integral.

The final inequality regarding the proper Darboux integral of $f$ follows immediately from:

$\ds \int_a^b \map f x \rd x = \overline {\int_a^b} \map f x \rd x = \underline {\int_a^b} \map f x \rd x$

whenever $f$ is Darboux integrable.

$\blacksquare$

This entry was named for Jean-Gaston Darboux.

1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.4$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Darboux's theorem
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integrability
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Darboux's theorem
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integrability