3.21 Theorem. Suppose that E is a nonempty subset of R, that a = E, and that f: E R. Then the following statements are equivalent: i) f is continuous at a Є E. ii) If x, converges to a and x₁ = E, then f(x) → f (a) as n→ ∞. 84 Chapter 3 Functions on R In particular, √√x is continuous on I = [0, ∞) by Exercise 2.2.5. By combining Theorem 3.21 with Theorem 2.12, we obtain the following result.

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