Next: 3.2.11 Monotonically Decreasing Functions Up: 3.2 Constant Interval Arithmetic Previous: 3.2.9 Examples with Piecewise Constant

## 3.2.10 Monotonically Increasing Functions

We will determine for any monotonically increasing function . Since g is monotontically increasing, . We assume that . We further assume that , so we may take . A simple proof by contradiction, which follows, shows that is an upper bound for g:

Assume that there is a point such that . Let , so . Furthermore, and imply that .

A quick review of the chart reveals that this situation is impossible. There is no such that since , , and .

The two assumptions made do not overly restrict the applicability of the proof. If , consider in place of g. If , consider in place of g, such that g' is monotonically increasing. If exists, it may be taken for y; otherwise, a trivial upper bound may be used.

Next: 3.2.11 Monotonically Decreasing Functions Up: 3.2 Constant Interval Arithmetic Previous: 3.2.9 Examples with Piecewise Constant
 Jeff Tupper March 1996