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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.

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.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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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.
Transcribed Image Text: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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