Constant function

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Definition

Let [ilmath]A[/ilmath] and [ilmath]B[/ilmath] be sets and [ilmath]f:A\rightarrow B[/ilmath] be any function between them. We call [ilmath]f[/ilmath] a "constant function" if:

• [ilmath]\exists b\in B\forall a\in A[f(a)=b][/ilmath]
  • In words: every [ilmath]a\in A[/ilmath] is sent to the same [ilmath]b\in B[/ilmath] by [ilmath]f[/ilmath].

We may write [ilmath]f[/ilmath] as:

• [ilmath]f:A\rightarrow B[/ilmath] given by [ilmath]f:a\mapsto b[/ilmath]

Terminology

• Constant map and constant mapping are obviously also used.
• Trivial map might be used to describe a constant map (and I am sure I have seen it) but this is rare and ill-advised.

References

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This terminology is... A-level, possibly earlier, and doesn't really need a page. So finding references will be difficult anyway as who would bother to write it!