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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 tex2html_wrap_inline34159 for any monotonically increasing function tex2html_wrap_inline34165 . Since g is monotontically increasing, tex2html_wrap_inline34661 . We assume that tex2html_wrap_inline34663 . We further assume that tex2html_wrap_inline34665 , so we may take tex2html_wrap_inline34667 . A simple proof by contradiction, which follows, shows that tex2html_wrap_inline34669 is an upper bound for g:

math13195

Assume that there is a point tex2html_wrap_inline34333 such that tex2html_wrap_inline34675 . Let tex2html_wrap_inline34677 , so tex2html_wrap_inline34339 . Furthermore, tex2html_wrap_inline34251 and tex2html_wrap_inline34661 imply that tex2html_wrap_inline34231 .

figure13201

A quick review of the tex2html_wrap_inline34257 chart reveals that this situation is impossible. There is no tex2html_wrap_inline34333 such that tex2html_wrap_inline34675 since tex2html_wrap_inline34231 , tex2html_wrap_inline34667 , and tex2html_wrap_inline34715 .

The two assumptions made do not overly restrict the applicability of the proof. If tex2html_wrap_inline34717 , consider tex2html_wrap_inline34719 in place of g. If tex2html_wrap_inline34723 , consider tex2html_wrap_inline34725 in place of g, such that g' is monotonically increasing. If tex2html_wrap_inline34731 exists, it may be taken for y; otherwise, a trivial upper bound may be used.

next up previous notation contents
Next: 3.2.11 Monotonically Decreasing Functions Up: 3.2 Constant Interval Arithmetic Previous: 3.2.9 Examples with Piecewise Constant
Jeff TupperMarch 1996