Lower semicomputable function

A function $f:X\to \mathbf{R}$ is lower semicomputable iff there exists a computable function $g:X\times \mathbf{N}\to \mathbf{Q}$ such that:

• for all $x\in X$ and all natural numbers $n$, we have $g(x,n+1)\ge g(x,n)$
• for all $x\in X$ we have $\underset{n\to \mathrm{\infty }}{lim}g(x,n)=f(x)$

The way to think of this is that given some fixed $x\in X$, the values $g(x,0),g(x,1),g(x,2),\dots$ are successive approximations of the value $f(x)$, and we have $g(x,0)\le g(x,1)\le g(x,2)\le \cdots \le f(x)$.