I am having trouble understanding the meaning of point functions. I know the mathematical definition but i don't think that i truly understand there true meaning.

Point functions:

Suppose $(X,d)$ is a metric space and $z∈X$. Then a nonnegative real function

$x \rightarrow d(x,z)$ defined on $X$ is a point function at $z$.

I have tried looking for examples w/ no luck.

EDIT: Here is some more context to my question.

http://s14.postimg.org/i1xzylwb5/0111152201.png