Suppose I wanted a function $f(x)$ such that the following properties are had.
$f(x)$ maps $\mathbb{R}\to\mathbb{R}$.
$f(a)>f(b)$ if $a>b$.
The function may or may not be continuous, but it doesn't have singularities.
Is there a special name for this type of function and does it have any special properties? (properties for all functions that meet the requirements.)
strictly monotone increasing. Any such function is continuous almost everywhere. But what do you mean by singularities? It could have jump discontinuities, in fact, it could have infinitely many. As pointed out by Merlinsbeard, it is in fact almost everywhere differentiable (which is even stronger)