Increasing function, decreasing function

An intuitive idea

A function f describes how a quantity changes. We say that f is increasing on an interval I if, as x increases, f(x) also increases. We say that it is decreasing if, as x increases, f(x) decreases.

Strict definition

Let f be a function defined on an interval I.

• f is increasing on I if, for all a and b in I: if a ≤ b, then f(a) ≤ f(b).
• f is decreasing on I if, for all a and b in I: if a ≤ b, then f(a) ≥ f(b).

If the inequalities are strict (f(a) < f(b)), we speak of strict increase or decrease. It is this strict version that is most useful for studying a function in detail.

Practical example

Let us consider f(x) = x² on the interval [0; +∞[. If we take a = 2 and b = 3 (so a <= b), we have f(2) = 4 and f(3) = 9, so f(a) <= f(b): the function is increasing on this interval.

But be careful: on the interval ]–∞; 0], the same function is decreasing: with a = –3 and b = –2 (a ≤ b), we get f(a) = 9 and f(b) = 4, so f(a) ≥ f(b).

A classic pitfall

The direction of variation always depends on the interval under consideration. To say “f(x) = x² is increasing” without specifying the interval is incomplete, or even incorrect, as this is only true on [0; +∞[.

Interval	Behaviour of x²
]–∞; 0]	decreasing
[0; +∞[	increasing

Another pitfall: a function may be constant over part of its domain (for example, f(x) = 3 for all x). In this case, it is neither strictly increasing nor strictly decreasing; it is constant.