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Exercise 5.2.5. Let fa(x) = { xa if x > 0 0 if x <0. (a) For which values of a is f continuous at zero? (b) For which values of a is ƒ differentiable at zero? In this case, is the derivative function continuous? (c) For which values of a is f twice-differentiable? Theorem 4.3.2 (Characterizations of Continuity). Let f AR, and let CE A. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e> 0, there exists a 8> 0 such that \x-c\ <8 (and x E A) implies |f(x) = f(c)| < €; - (ii) For all V.(f(c)), there exists a Vs (c) with the property that x = V&(c) (and TEA) implies f(x) = V. (f(c)); (iii) If (xn) → c (with xn E A), then f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). X-C

Exercise 5.2.5. Let fa(x) = { xa if x > 0 0 if x <0. (a) For which values of a is f continuous at zero? (b) For which values of a is ƒ differentiable at zero? In this case, is the derivative function continuous? (c) For which values of a is f twice-differentiable? Theorem 4.3.2 (Characterizations of Continuity). Let f AR, and let CE A. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e> 0, there exists a 8> 0 such that \x-c\ <8 (and x E A) implies |f(x) = f(c)| < €; - (ii) For all V.(f(c)), there exists a Vs (c) with the property that x = V&(c) (and TEA) implies f(x) = V. (f(c)); (iii) If (xn) → c (with xn E A), then f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). X-C

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 36E
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Exercise 5.2.5. Let fa(x) = { xa if x > 0 0 if x <0. (a) For which values of a is f continuous at zero? (b) For which values of a is ƒ differentiable at zero? In this case, is the derivative function continuous? (c) For which values of a is f twice-differentiable?
Transcribed Image Text:Exercise 5.2.5. Let fa(x) = { xa if x > 0 0 if x <0. (a) For which values of a is f continuous at zero? (b) For which values of a is ƒ differentiable at zero? In this case, is the derivative function continuous? (c) For which values of a is f twice-differentiable?
Theorem 4.3.2 (Characterizations of Continuity). Let f AR, and let CE A. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e> 0, there exists a 8> 0 such that \x-c\ <8 (and x E A) implies |f(x) = f(c)| < €; - (ii) For all V.(f(c)), there exists a Vs (c) with the property that x = V&(c) (and TEA) implies f(x) = V. (f(c)); (iii) If (xn) → c (with xn E A), then f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). X-C
Transcribed Image Text:Theorem 4.3.2 (Characterizations of Continuity). Let f AR, and let CE A. The function f is continuous at c if and only if any one of the following three conditions is met: (i) For all e> 0, there exists a 8> 0 such that \x-c\ <8 (and x E A) implies |f(x) = f(c)| < €; - (ii) For all V.(f(c)), there exists a Vs (c) with the property that x = V&(c) (and TEA) implies f(x) = V. (f(c)); (iii) If (xn) → c (with xn E A), then f(xn) → f(c). If c is a limit point of A, then the above conditions are equivalent to (iv) lim f(x) = f(c). X-C
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