What do these actually mean ? 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 $z \in X$. Then a non negative real function
$x$->$d(x,z)$ defined on $X$ is a point function at z.
Point Like functions:
Suppose $(X,d)$ is a non empty metric space and $u:X->R+$.Then u is a pointlike function on $X$ if, and only if, $u(a)-u(b)<=d(a,b)<=u(a)+u(b)$ for all $a,b \in X$.
Any help would be appreciated.
What you call point function is usually known as distance function (of a point). In a Euclidean space, its graph is a cone, and the function happens to be convex. The convexity of distance function turns out to reflect the curvature of an underlying space. For example, a Riemannian manifold is "globally nonpositively curved" (technically speaking, is a CAT(0) space) if and only if the distance function is convex. Simple example: on the sphere, which does have positive curvature, the distance function is not convex, it even has a point of maximum. Since the distance function is available on every metric space, its convexity properties lead to concepts of curvature bounds that are applicable to somewhat rough metric spaces (think of the surface of a cube).
I can't comment on pointlike (distance-like?) functions; the definition you gave makes perfect sense (it's a function that mimics the behavior of $z\mapsto d(x,z)$) but I haven't seen it used yet. Maybe if you included the source of the definition...
