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, thenthe 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 byg(x)= x sin (1/x),x * 0,= 0,0,%3Dis continuous at every point.15.J. Let h be defined for x # 0, x € R, byh(x)sin (1/2),* * 0.Show that no matter how h is defined at r =0, it will be discontinuous at r == 0.

Answered: Image /qna-images/answer/64b4c8a9-04d6-4594-ba5e-55ebe6557f50.jpg

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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