(Real analysis, topology):

We need to know a couple facts about continuous functions and dense subsets, first.

1. From topology, in a metric space, a dense subset is such that every point of the space is either
   1. part of the dense subset, or
   2. the limit of a sequence of points from the dense subset
2. Continuous functions commute with limits: lim_n f(x_n) = f(lim_n x_n)

For concreteness' sake, let's start with continuous, real-valued functions, f:R→R. Let D be a dense subset of R. If g is another continuous function that is the same as f on D, then it turns out that f is the same as g everywhere.

Why should such a thing be true? We already know f(x) = g(x) when x is in D, so we need to look at what happens outside of D. By fact (1), for all real numbers a outside of D, there is a sequence x_n contained entirely in D such that lim_n x_n = a. Now we already know how f and g act on x_n; since x_n is contained in D, f(x_n) = g(x_n).

However, by fact (2)

lim_n f(x_n) = f(lim_n x_n) = f(a)

lim_n g(x_n) = g(lim_n x_n) = g(a)

and the right-hand sides are equal because the left-hand sides are equal. So f(a) = g(a) for any a outside of D, and f(x) = g(x) for any x inside of D, so f and g are equal everywhere.

In particular, if you define a real function only on the rational numbers, you can uniquely extend it to a function on all of the reals by writing an irrational number as the limit of a sequence of rational numbers, and defining the function on the irrational numbers as the limit of the image of this sequence. You also would need to show that all of these limits exist, but that is another story.

In fact, while we did do this in a metric space, a very similar thing can be done for general topological spaces, by considering the topological definition of dense subsets.