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Unformatted text preview: Math 410 HW 1 Solution:s January 24, 2009 1. Consider the function f defined on the unit interval [0 , 1] by f ( x ) = if x = 0 x sin(1 /x ) if 0 < x ≤ 1. (a) Is f continuous on [0 , 1]? Solution: The function f as defined on the halfopen interval (0 , 1] is obviously continuous, being the composition and product of continu ous welldefined functions. We therefore only need to check continuity of f at x = 0. Note that x ≤ x sin(1 /x ) ≤ x so that the squeeze theorem implies lim x → + f ( x ) = 0 = f (0) . This guarantees that f is also continuous at x = 0. Thus, f is indeed continuous on [0 , 1]. (b) Does f belong to C 1 [0 , 1]? Solution: Standard calculus reveals that for x in the open interval (0 , 1), f ( x ) = 1 x cos 1 x + sin 1 x , which is continuous. However, there is no possible way to guess the value of f (0) without doing some work. Using the definition of the derivative, f (0) = lim h → + f (0 + h ) f (0) h = lim h → + f ( h ) h = lim h → + h sin(1 /h ) h = lim h → + sin(1 /h ) , which does not exist. Hence, f does not have a derivative (much less a continuous derivative) at x = 0, so f does not belong to C 1 [0 , 1]. 1 2. Find a function g ∈ C ∞ ( R ) which satisfies g ( x ) = if x ≤  1 1 if x ≥ 0....
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 Math, Calculus, Topology, Derivative, 1 2K

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