Inverse Image of Convex Set under Monotone Mapping is Convex

Theorem
Let $(X,\le)$ and $(Y, \preceq)$ be ordered sets.

Let $f:X \to Y$ be a monotone mapping.

Let $C$ be a convex subset of $Y$.

Then $f^{-1}(C)$ is convex in $X$.

Proof
Suppose $f$ is increasing.

Let $a,b,c \in X$ with $a \le b \le c$ and let $a,c \in f^{-1}(C)$.

Thus by the definition of inverse image, $f(a), f(c) \in C$.

By the definition of an increasing mapping, $f(a) \preceq f(b) \preceq f(c)$.

Thus by the definition of a convex set, $f(b) \in C$.

Then by the definition of inverse image, $b\in f^{-1}(C)$.

Since this holds for all such triples, $f^{-1}(C)$ is convex.

A similar argument applies if $f$ is decreasing.