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Next: 3.2.12 Lower Bounds Up: 3.2 Constant Interval Arithmetic Previous: 3.2.10 Monotonically Increasing Functions

3.2.11 Monotonically Decreasing Functions

 

We will determine tex2html_wrap_inline34159 for any monotonically decreasing function tex2html_wrap_inline34165 . Since g is monotontically decreasing, tex2html_wrap_inline34741 . We assume that tex2html_wrap_inline34663 , and that tex2html_wrap_inline34745 , where tex2html_wrap_inline34747 ;   tex2html_wrap_inline34749 , so that tex2html_wrap_inline34751 . Take tex2html_wrap_inline34753 ; 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_inline34741 imply that tex2html_wrap_inline34227 .

figure13367

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_inline34227 , tex2html_wrap_inline34753 , and tex2html_wrap_inline34801 .

next up previous notation contents
Next: 3.2.12 Lower Bounds Up: 3.2 Constant Interval Arithmetic Previous: 3.2.10 Monotonically Increasing Functions
Jeff TupperMarch 1996