Since the Completeness Property demands

$$L \left( f, \ P \right) \le \sup_P L \left( f, \ P \right) \le \inf_Q U \left( f, \ Q \right) \le U \left( f, \ Q \right)$$

the following definition for the Darboux Integral can be given. If \( f : \left[ a, \ b \right] \rightarrow \mathbb{R} \) is a bounded function then the Upper Darboux Integral is

$$\overline{ \int_a^b } f = \inf_Q U \left( f, \ Q \right)$$

Similarly, the Lower Darboux Integral is

$$\underline{ \int_a^b } f = \sup_P L \left( f, \ P \right)$$

Moreover, by definition

$$\underline{ \int_a^b } f \le \overline{ \int_a^b } f$$

Let \( f : \left[ a, \ b \right] \rightarrow \mathbb{R} \) be a bounded function on \( \left[ a, \ b \right] \). If

$$\overline{ \int_a^b } f = \underline{ \int_a^b } f$$

then the Darboux integral is

$$\int_a^b f = \overline{ \int_a^b } f = \underline{ \int_a^b } f$$

It is the case that \( f \) is Riemann integrable. Skip to the section on Riemann Sums for further discussion.