A function where are totally ordered sets is said to be increasing if for every elements , implies .
A simple example of an increasing function is the map defined by . In fact, it's easy to see that the identity map is an increasing map on any totally ordered set.
If implies we say that is strictly increasing.
The notion of an increasing function is generalized to the context of partially ordered sets by order-preserving functions.
If the function is differentiable then is increasing if and only if for all . If for all then is strictly increasing, but the converse does not hold: for example, the function is strictly increasing on the interval , but .
If the domain of the function is the integers (or the positive integers or nonnegative integers) then is increasing (or strictly increasing) if and only if the sequence of its values is an increasing sequence (respectively, strictly increasing sequence).