Definition:Negative Part/Also defined as/Real-Valued Function

From ProofWiki
Jump to navigation Jump to search

Negative Part: Also defined as

Some sources insist, when defining the negative part, that $f$ be a real-valued function:

$f: X \to \R$

That is, that the codomain of $f$ includes neither the positive infinity $+\infty$ nor the negative infinity $-\infty$.


However, $\R \subseteq \overline \R$ by definition of $\overline \R$.

Thus, the main definition as provided on $\mathsf{Pr} \infty \mathsf{fWiki}$ incorporates this approach.


Hence it is still the case that:

$\forall x \in X: \map {f^-} x := \begin {cases} -\map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$


Also see

  • Results about negative parts can be found here.


Sources


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Negative_Part/Also_defined_as/Real-Valued_Function&oldid=616979"