Monotonically Increasing

Let f be real on (a,b). f is monotone increasing if on (a,b),
a< x < y <b implies f(x)less then or equal to f(y)

Let f: A -> B, where A and B are totally ordered. Then f is said to be monotonically increasing if, whenever a₁ \in A < a₂ \in A, f(a₁) <= f(a₂). If the inequality is strict, f is said to be strictly monotonically increasing.