Definition:Stieltjes Function

Definition
Let $f: \R \to \overline{\R}$ be a mapping, where $\overline{\R}$ denotes the extended real numbers.

Then $f$ is said to be a Stieltjes function iff:


 * $(1): \quad f$ is increasing
 * $(2): \quad f$ is left-continuous.

Also known as
Some sources insist that the codomain of a Stieltjes function $f$ be $\R$.

That is, they exclude the possibility that $f$ assumes the values $\pm \infty$.

To express that a Stieltjes function $f$ does not assume infinite values, one may call $f$ a finite Stieltjes function.

Also see

 * Stieltjes Function of Measure on Real Numbers