A metric space is a very special case of a topological space, and the definition of a continuous function between 2 metric spaces is equivalent to the following:
A function f:X→Y between metric spaces X and Y is continuous at a point x∈X iff for all positive ε there exists a positive δ such that for any x', if d(x,x') < δ then d(f(x),f(x')) < ε. That is, if we stay sufficiently close to x, then our image through f stays close to f(x). If f is continuous on all points of a set U, then f is called continuous on U. And if f is continuous on all points of X, f is called simply continuous.
In non-standard analysis, we have the following definition:
A (standard!) function f:X→Y between metric spaces X and Y is continuous at a (standard!) point x∈X iff whenever the standard approximation of a (possibly non-standard) x' is x, the standard approximation of f(x') is f(x).
Continuity on a set continues to be defined as before.
Note that we have fewer quantifiers in the definition, making it possibly simpler. We also only need to consider an "infinitesimally small" neighbourhood of x, rather than the "arbitrarily small" neighbourhoods of x required in the "standard" definition.