Mapping is Constant iff Image is Singleton

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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 $f_c: S \to T$).

$\blacksquare$


Sources