Date: Around 1995
Please define what is meant by a "monotonic" function. Is it any function
with a constant second derivative, any function with a second derivative
that does not change sign, any function with third and higher derivatives
equal to zero, or what?
A function f(x) is monotonic increasing if a < b implies f(a) < f(b);
A function f(x) is monotonic decreasing if a < b implies f(a) > f(b).
There are also monotonic non-decreasing [a < b implies f(a) < or = f(b)]
and monotonic non-increasing [a < b implies f(a) > or = f(b)] functions.
The function need not be differentiable; if it is, equivalent definitions
can be given in terms of the sign of f'. f" and higher-order derivatives
are not relevant. Ordinarily, if a function is simply referred to as being
monotonic, that means the function is either monotonic increasing or else
monotonic decreasing. So (again, assuming differentiability) it means the
sign of f' does not change.
In looking through some of my books I find that some authors do not use the
non-decreasing and non-increasing distinctions given above. That is, the
definition I gave for "monotonic non-decreasing" they use to define "mono-
tonic increasing" and what I gave for "monotonic non-increasing" they use to
define "monotonic decreasing".
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Update: June 2012