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15.1. Using the results of the preceding exercise, show that the function g, defined on R to R by g(x) = x sin (1/x), x * 0, 0, 0, is continuous at every point. 15.J. Let h be defined for x 0, x e R, by h(x) sin (1/2), * * 0. Show that no matter how h is defined at x = 0, it will be discontinuous at x = 0.

15.1. Using the results of the preceding exercise, show that the function g, defined on R to R by g(x) = x sin (1/x), x * 0, 0, 0, is continuous at every point. 15.J. Let h be defined for x 0, x e R, by h(x) sin (1/2), * * 0. Show that no matter how h is defined at x = 0, it will be discontinuous at x = 0.

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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15.- Solve based on the theorem in photo 2, please

15.2. THEOREM. Let a be a point in the domain D of the function f. The following statements are eguivalent: (a) f is continuous at a. (b) If e is any positive real number, there exists a positive number ô (e) such that if x € D and x – al < 8(e), then \f(x) – f(a)| < e. (c) If (x,) is any sequence of elements of D which converges to a, then the sequence (f (xn)) converges to f(a).
Transcribed Image Text:15.2. THEOREM. Let a be a point in the domain D of the function f. The following statements are eguivalent: (a) f is continuous at a. (b) If e is any positive real number, there exists a positive number ô (e) such that if x € D and x – al < 8(e), then \f(x) – f(a)| < e. (c) If (x,) is any sequence of elements of D which converges to a, then the sequence (f (xn)) converges to f(a).
15.I. Using the results of the preceding exercise, show that the function g, defined on R to R by g(x) = x sin (1/x), x * 0, = 0, 0, %3D is continuous at every point. 15.J. Let h be defined for x # 0, x € R, by h(x) sin (1/2), * * 0. Show that no matter how h is defined at r = 0, it will be discontinuous at r = = 0.
Transcribed Image Text:15.I. Using the results of the preceding exercise, show that the function g, defined on R to R by g(x) = x sin (1/x), x * 0, = 0, 0, %3D is continuous at every point. 15.J. Let h be defined for x # 0, x € R, by h(x) sin (1/2), * * 0. Show that no matter how h is defined at r = 0, it will be discontinuous at r = = 0.
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