Calculus of One and Several Variables

4.1 Equivalent Formulations Of Continuity

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. You may find this equivalent description is much easier to use than the formal ε,δ definition given earlier. There is really no good reason for this, but in fact, it tends to be easier for most people to use although I don’t understand why this seems to be the case.

Proof: Suppose first that f is continuous at x and let x_n → x. Let ε > 0 be given. By continuity, there exists δ > 0 such that if

< δ,y ∈ D, then < ε. However, there exists n_δ such that if n ≥ n_δ, then < δ and so for all n this large,

|f (x)− f (x )| < ε n

which shows f

→ 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 x_n ∈ D

such that < , yet

|f (x)− f (xn)| ≥ ε.

But this is clearly a contradiction because, although x_n → x, f

fails to converge to f. It follows f must be continuous after all. ■

Proof: Since f

≤ l and f is continuous at x, it follows from Theorem 3.2.11 and Theorem 4.1.1,

f (x) = nli→m∞ f (xn ) ≤ l.

The other case is entirely similar. ■

The following is a useful characterization of a continuous function which ties together the above conditions. I am being purposely vague about the domain of the function and its range because this theorem is a general result which holds whenever it makes sense.