# Mapping is Constant iff Image is Singleton

## Theorem

A mapping is a constant mapping if and only if its image is a singleton.

## Proof

### Necessary Condition

Let $f_c: S \to T$ be a constant mapping. Then from the definition of the image of a element:

$\forall x \in S: \map {f_c} x = c \implies \Img S = \set c$

Thus the image of $f_c: S \to T$ is a singleton.

### Sufficient Condition

Let $f: S \to T$ such that its image is a singleton $\set c \subseteq T$.

 $\ds \forall x \in S: \map f x$ $\in$ $\ds \Img f$ Definition of Image of Mapping $\ds \leadsto \ \$ $\ds \forall x \in S: \map f x$ $\in$ $\ds \set c$ By Hypothesis $\ds \leadsto \ \$ $\ds \forall x \in S: \map f x$ $=$ $\ds c$ Definition of Singleton

Thus $f: S \to T$ is a constant mapping (and we can write it $f_c: S \to T$).

$\blacksquare$