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 $g(x,n+1)\geq g(x,n)$
• for all $x\in X$ we have $\lim _{n\to \infty }g(x,n)=f(x)$

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