A topological space (X, t) is an ordered pair of a set X (that is at least countable in most cases, but also we can look at finite sets if we want), together with a prescribed family of subsets on it called t (these subsets are called open sets and they also must have some nice properties like arbitrary unions and finite intersections of open sets must also be open sets, the empty set and the whole X are also members of t). This family of open sets t is called a topology.

By definition, a function f : (X, t)-->(Y, s) between topological spaces (X, t) and (Y, s) is continuous at a point p if each pre-image of an open set V containing the point f(p) is an open set U containing p. A function is continuous on a subset if it is continuous at every point p of that subset. The key point is that there cannot be continuous functions between bare sets, only between topological spaces.

So, if a subset is not an element of the topology, then saying that a function is continuous on that subset doesn't make sense.

In general, arbitrary subsets of a given set can be very nasty, most of them cannot even be defined.

By defining a topology on a set, you say exactly what kind of subsets of that set you will be looking at, and what we want from these open sets is to behave nicely (an arbitrary union of open sets must be an open set, and finite intersection of them also must be an open set).

The easiest way to define a topology for a subset A of X that is not a member of the topology t is by defining a relative topology on it (taking intersections of A with all members of t). Also, you can try and prescribe your own topology on this subset (in some way) but it is rather uncommon for such a topology *not* to be inherited from the whole space.