Mapping is Constant iff Image is Singleton

Theorem
A mapping is a constant mapping iff its image is a singleton.

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


 * $\forall x \in S: f_c \left({x}\right) = c \implies \operatorname{Im} \left({S}\right) = \left\{{c}\right\}$

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

Necessary Condition
Let $f: S \to T$ such that its image is a singleton $\left\{{c}\right\} \subseteq T$.

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