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3.2.4 Constant Functions

Consider the constant function tex2html_wrap_inline32831 . Since g is constant, tex2html_wrap_inline34325 . Take any tex2html_wrap_inline34327 ; a simple proof by contradiction, which follows, shows that tex2html_wrap_inline34329 is an exact bound of g:

math12320

Assume there is a point tex2html_wrap_inline34333 such that tex2html_wrap_inline34335 . Let tex2html_wrap_inline34273 , so tex2html_wrap_inline34339 :

eqnarray12325

Furthermore, tex2html_wrap_inline34251 and tex2html_wrap_inline34325 together imply that tex2html_wrap_inline34229 .

figure12329

A quick review of the tex2html_wrap_inline34257 chart reveals this situation is impossible, since tex2html_wrap_inline34229 implies that tex2html_wrap_inline34301 . The tex2html_wrap_inline34257 chart predicts the sign of tex2html_wrap_inline34221 since tex2html_wrap_inline34273 .

So, for any tex2html_wrap_inline34327 , tex2html_wrap_inline34377 . It follows that tex2html_wrap_inline34329 is a lower and upper bound for g, over tex2html_wrap_inline34383 :

math12403

The intuitive interval model originally given is now seen to be correct:

math12410

since it is equivalent to:

math12417

assuming that g is total.

next up previous notation contents
Next: 3.2.5 Optimality Up: 3.2 Constant Interval Arithmetic Previous: 3.2.3 Charts
Jeff TupperMarch 1996