Just as in the case of a function of one variable, there is a very useful way of thinking of continuity in terms of limits of sequences found in the following theorem. In words, it says a function is continuous if it takes convergent sequences to convergent sequences whenever possible.
Proof: Suppose first that f is continuous at x and let xn→ x. Let ε > 0 be given. By continuity, there exists δ > 0 such that if
which shows f
Now suppose the condition about taking convergent sequences to convergent sequences holds at x. Suppose f fails to be continuous at x. Then there exists ε > 0 and xn ∈ D
But this is clearly a contradiction because, although xn→ x, f
You can replace ℝp, ℝq with arbitrary normed linear spaces with no change. You simply replace