Next let s be a sequence in a metric space. Recall the definitions of cluster point and limit point when dealing with a sequence. These definitions have equivalent formulations in a metric space.
A sequence converges to p if every ε has an index n such that the entire sequence, beyond index n, is within ε of p. This is the same as saying every open set containing p contains most of s.
Similarly, p is a cluster point if every ε has infinitely many points of s within ε of p. This is almost the same as saying p is a limit point of the set s. If s does not contain p, they are the same. Keep shrinking ε to find new points, closer and closer to p. This builds a sequence headed towards p. Of course if s contains p then p is a limit point of s, but need not be a cluster point.
We can now define continuity in terms of limits. This is probably the definition you've seen before; the definition you are use to. As you might surmise, it coincides with the topological definition. Let f be a function from one metric space into another, and let f(p) = q. Recall that continuity means every open set about q implies an open set about p whose image is wholly contained in the open set about q. Open sets are now open balls with specific radii, so here is the same definition. For every ε there is δ, such that |p,x| < δ implies |q,f(x)| < ε.
Here is yet another equivalent definition of continuity. The function f is continuous at p if every sequence s in the domain with limit p maps to a sequence t in the range with limit q. Let's see why this is really the same as the δ ε definition.
Assume the distance restriction is in force, and let s be a sequence that approaches p. Let t be the image sequence and suppose it does not approach q. Let ε be the radius of a ball about q that does not contain most of t. That is, the sequence t lies outside the ε ball infinitely often. However, the open ball of radius δ about p contains most of s, since s appraoaches p, and that means most of t actually lies in the ε ball about q.
Conversely, let every sequence with limit p map to a sequence with limit q, and suppose an ε ball about q does not have a δ ball about p in its preimage. Step δ through the reciprocols of the integers. At each step we find at least one point within δ of p whose image is more than ε away from q. These points form a sequence that converges to p, yet the image sequence does not converge to q. This is a contradiction, hence there is a δ-ball in the preimage of every ε-ball. The open set definition, the δ ε definition, and the convergent sequence definition are all equivalent.
As before, f is continuous throughout a region if it is continuous at every point in the region.