I learnt increasing functions as x < y  ⇒ f(x) ≤ f(y).
ie. A constant function counts as increasing, for example.

I learnt the definition on the page as "strictly increasing".

Probably different people use different definitions though.

Good points, thank you.

I have redone Increasing Functions

Please check it out and let me know if it passes.

According to the new definition, a constant function is increasing and decreasing, rather than neither.

Also, I'd say something about how for local extrema, the point should be "in" the interval, rather than on the edge.

For example, with f(x) = x on the interval [1,2], the maximum there would be at x=2. But you wouldn't consider that a local maximum of the function.

Well, that problem only happens when you work with a specific interval. So maybe the page should keep the "f(a) ≥ f(x) for all x in the interval", but say it has to be true for all small enough intervals rather than just one.

Or is that complicating things too much?

About constant functions …

A Constant Function neither increases nor decreases:

You realize that this is wrong? In fact, a Constant Function is both increasing and decreasing (but not strictly) along its whole domain.

Think about it.

Last edited by JaneFairfax (2009-09-16 11:29:23)

According to the definition, a function y=f(x) is increasing iff

when x1 < x2 then f(x1) ≤ f(x2)

Hence, if y=f(x) is non-increasing, then there exist x1 < x2 such that f(x1) > f(x2). But a constant function cannot possibly have f(x1) > f(x2). Therefore a constant function cannot be non-increasing.

By a similar argument a constant function cannot be non-decreasing either. Therefore it must be both increasing and decreasing.

Last edited by JaneFairfax (2009-09-16 15:23:45)

I think what you meant was that a constant function is neither strictly increasing nor strictly decreasing.

Well, aren’t constant functions just fascinating? They are both increasing and decreasing, yet neither strictly increasing nor strictly decreasing. Amazing how we take so many things for granted, only to find, on closer inspection, that we have been mistaken all along.