Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, $a+2h$,....$a+(n-1)h$ are rational numbers. Consider P be a set of all these points. As $n$ goes to infinity the cardinality of the set P goes to infinity. The interval is uncountably infinite whereas the set P is countable.
Question is for large n, can we say the set P is an approximation of the interval?
Basically I have a function which need to be defined over an interval. But I would like to define the map only on the set $P$. Does that ensure the map would be well defined (almost approximately) over the interval?