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
.
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.
Jeff Tupper | March 1996 |