Consider the real-valued function of one variable f: R → R defined by - {₁+2²+ (a) Show that f'(0) f(t)= = for t= 0 +2t² sin () for t = 0. = 1, and that the derivative f'(t) exists and is bounded in the open interval (-1,1). (b) Show that f'(t) is not continuous at t = 0. (c) Show that f is not one-to-one in any neighborhood of 0 (and hence not invertible there). [This shows that, even in the one-dimensional case n = 1, the continuity of the deriva- tive f' at the point a is needed in the Inverse Function Theorem, even if f'(a) #0!]
Consider the real-valued function of one variable f: R → R defined by - {₁+2²+ (a) Show that f'(0) f(t)= = for t= 0 +2t² sin () for t = 0. = 1, and that the derivative f'(t) exists and is bounded in the open interval (-1,1). (b) Show that f'(t) is not continuous at t = 0. (c) Show that f is not one-to-one in any neighborhood of 0 (and hence not invertible there). [This shows that, even in the one-dimensional case n = 1, the continuity of the deriva- tive f' at the point a is needed in the Inverse Function Theorem, even if f'(a) #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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