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The following is a definition of a point function I came across in Metric Spaces by Michael O'Searcoid :

Definition: Suppose $(X,d)$ is a metric space and $z \in X$. We shall call the non-negative real function $x \to d(z,x)$ defined on $X$ the point function at $z$ and denote it by $\delta_z$.

I am having trouble understanding what this definition is telling me.

What I think it is saying is that there is a function that just looks at the difference between $z$ and every value of $x \in X$.

I would appreciate it if someone could provide some clarification on this definition, and possibly provide an example.

GovEcon
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  • I guess that it is like a 'new origin', so you compute the distance from $x$ to it, that is, to $z$. – Sigur Jun 25 '13 at 01:35
  • Have you encountered the term "point function" and this definition in a book? What this function does is clear. But I have never heard the term. – Julien Jun 25 '13 at 01:36
  • @julien Yes I did, I forgot to include the reference. I will add it to the question now. That also explains why Google didn't have the answer. – GovEcon Jun 25 '13 at 01:36
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    The only thing I can think of about $\delta_z$ in such a generality is that it is $1$-Lipschitz by triangular inequality. – Julien Jun 25 '13 at 01:47

1 Answers1

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Yes, you are correct.

A simple example would be the real numbers with the Euclidean metric. Then let $z = 0$ and the point function sends $x \rightarrow |x|$.

Benjamin Dickman
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