Help with exercise 19.CProve by definition 19.1 DEFINITION. If c is a cluster point of D and belongs to D,we say that a real number L is the derivative of f at c if for every positivenumber e there is a positive number 8(e) such that if x belongs to D andif 0 < |æ – cl < ô(e), thenf(x) – f(c)(19.1)L<x - cIn this case we write f' (c) for L.Alternatively, we could define f' (c) as the limit(19.2)f(x) - f(c)lim(x € D).x - cIt is to be noted that if c is an interior point of D, then in (19.1) weconsider the points x both to the left and the right of the point c. Onthe other hand, if D is an interval and c is the left end point of D, thenin relation (19.1) we can only take x to the right of c. In this case wesometimes say that "L is the right-hand derivative of f at r = c." How-ever, for our purposes it is not necessary to introduce such terminology.Whenever the derivative of f at c exists, we denote its value by f' (c).In this way we obtain a function f' whose domain is a subset of thedomain of f. We now show that continuity of f at c is a necessary con-dition for the existence of the derivative at c. 19.B. If f and g are real-valued functions defined on an interval J, and if theyare differentiable at a point e, show that their product h, defined by h(x)f(x)g(x), x € J, is differentiable at c andh' (c) = f'(c)g(c) + f(c)gʻ(c).19.C. Show that the function defined for x # 0 byf(x)sin (1/x)is differentiable at each non-zero real number. Show that its derivative is notbounded on a neighborhood of x =identities, the continuity of the sine and cosine functions, and the elementarylimiting relation0. (You may make use of trigonometricsin uas u → 0.)

Name: Help with exercise 19.CProve by definition 19.1 DEFINITION. If c is a
Uploaded: 2024-03-05T11:53:49.000Z
Channel: Bartleby
Description: Help with exercise 19.CProve by definition 19.1 DEFINITION. If c is a cluster point of D and belongs to D,we say that a real number L is the derivative of f at c if for every positivenumber e there is a positive number 8(e) such that if x belongs to