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 $$f_c: S \to T$$).