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up vote 0 down vote favorite Suppose that $A$ and $B$ are nonempty subsets of $mathbb{R}$ , and that $f : A rightarrow B$ is a function satisfying $forall: a_1, a_2 in A,:a_1 leq a_2 implies f(a_1) leq f(a_2)$ . (Such a function is said to be weakly increasing.) Furthermore, suppose that $f$ is surjective (i) Prove that if $M$ is a largest element of $A$ , then $f(M)$ is a largest element of $B$ . (ii) Why is the assumption that $f$ is surjective necessary? Give an example of nonempty subsets $A$ and $B$ of $mathbb{R}$ , a weakly increasing function $f : A rightarrow B$ , and a largest element $M$ of $A$ such that $f(M)$ is not a largest element of $B$ . real-analysis functions share | cite | improve this qu