Inverse Image of Convex Set under Monotone Mapping is Convex

Theorem
Let $\left({X,\le}\right)$ and $\left({Y, \preceq}\right)$ 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.