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Problem 9.2.13. Prove Theorem 9.2.12 Theorem 9.2.12. Suppose f is continuous at a and g is continuous at f(a). Then g of is continuous at a. (Note that (go f) (x) = g(f(x)),) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1.

Problem 9.2.13. Prove Theorem 9.2.12 Theorem 9.2.12. Suppose f is continuous at a and g is continuous at f(a). Then g of is continuous at a. (Note that (go f) (x) = g(f(x)),) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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12:41 M A personal.psu.edu/ecb! + 4 (Note that (g o f)(x) = g(f(x)).) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1. Theorem 9.2.1. The function f is continuous at a if and only if f satisfies the following property: sequences (xn), if lim xn = a then lir A in-context The above theorems allow us to build continuous functions from other continuous functions. For example, knowing that f(x) = x and g(x) = c are continuous, we can conclude that any polynomial, p(æ) n-1 anx" + an-1x"- + a1x + ao is continuous as well. We also know that functions such as f(x) = sin (e") are continuous without having to rely on the definition. II II
Transcribed Image Text:12:41 M A personal.psu.edu/ecb! + 4 (Note that (g o f)(x) = g(f(x)).) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1. Theorem 9.2.1. The function f is continuous at a if and only if f satisfies the following property: sequences (xn), if lim xn = a then lir A in-context The above theorems allow us to build continuous functions from other continuous functions. For example, knowing that f(x) = x and g(x) = c are continuous, we can conclude that any polynomial, p(æ) n-1 anx" + an-1x"- + a1x + ao is continuous as well. We also know that functions such as f(x) = sin (e") are continuous without having to rely on the definition. II II
12:41 Problem 9.2.13. Prove Theorem 9.2.12 Theorem 9.2.12. Suppose f is continuous at a and g is continuous at f(a). Then go f is continuous at a. (Note that (go f)(x) = g(f(x)) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1. Theorem 9.2.1. The function f is continuous at a if and only if f satisfies the following property: V sequences (xn), if lim xn = a then lir n 0 in-context The above theorems allow us to build continuous functions from other continuous functions. For example, knowing that f(x) continuous, we can conclude that any polynomial, = x and g(x) = c are p(x) = a,a" + an-1x" n-1 + + ajx + ao II II
Transcribed Image Text:12:41 Problem 9.2.13. Prove Theorem 9.2.12 Theorem 9.2.12. Suppose f is continuous at a and g is continuous at f(a). Then go f is continuous at a. (Note that (go f)(x) = g(f(x)) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1. Theorem 9.2.1. The function f is continuous at a if and only if f satisfies the following property: V sequences (xn), if lim xn = a then lir n 0 in-context The above theorems allow us to build continuous functions from other continuous functions. For example, knowing that f(x) continuous, we can conclude that any polynomial, = x and g(x) = c are p(x) = a,a" + an-1x" n-1 + + ajx + ao II II
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