Increasing function

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An increasing function (sometimes called a strictly increasing function) is a function f satisfying the following condition for any two distinct elements x_1, x_2 in its domain:

x_1 < x_2 \implies f(x_1) < f(x_2)

Note: Sometimes the term increasing is used for a non-decreasing function, where we do not require strict inequality but only insist on f(x_1) \le f(x_2).

Also note that for functions whose domain is empty or a single point, the definition of being increasing is vacuously satisfied and is uninteresting.