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Name: mortis
Status: N/A
Age: N/A
Location: N/A
Country: N/A
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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