HW 6 Solutions

HW 6 Solutions - Math 117 F 09 HW#6(sample solutions P.79#12 The symbol | f | denotes the function whose outputs are the absolute values of the

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Unformatted text preview: Math 117, F 09, HW#6 (sample solutions) P.79, #12. The symbol | f | denotes the function whose outputs are the absolute values of the outputs of f , that is, | f | : D → R and | f | ( x ) = | f ( x ) | for all x ∈ D . We will need an identity about differences of absolute values. This is part (iv) of Theorem 0.5: | a | - | b | ≤ | a- b | ( † ) for any real numbers a and b . (If you have not seen this identity before, you should read the proof, or, better yet, make your own proof.) Now set L = lim x → x f ( x ); we want to prove that lim x → x | f ( x ) | = | L | . So let > 0. Since lim x → x f ( x ) = L , there exists δ > 0 such that | f ( x )- L | < for all x ∈ D with 0 < | x- x | < δ . By the identity ( † ), we then get | f ( x ) | - | L | ≤ | f ( x )- L | < for all x ∈ D with 0 < | x- x | < δ . This proves that | L | = lim x → x | f ( x ) | = lim x → x | f | ( x ). P.79, #13. The graph of this function is a “sawtooth”, which makes a jump at each integer. Note that for any integer n , the points x in the half-open interval [ n,n +1) satisfy [ x ] = n . Thus, f ( x ) = x- n for all x ∈ [ n,n + 1). Claim : f has a limit at a number x ∈ R if and only if x is not an integer. First suppose that x is not an integer. Set n = [ x ]; then x ∈ ( n, n + 1). Set δ 1 = min { x- n, n + 1- x } , so that δ 1 > 0 and ( x- δ 1 , x + δ 1 ) ⊆ ( n, n + 1). In particular, f ( x ) = x- n for all x ∈ ( x- δ 1 , x + δ 1 ). We claim that lim x → x f ( x ) = x- n . Now let > 0. Set δ = min { δ 1 , } . For all x ∈ R with 0 < | x- x | < δ , we have x ∈ ( x- δ 1 , x + δ 1 ) and f ( x ) = x- n , and so | f ( x )- ( x- n ) | = | ( x- n )- ( x- n ) | = | x- x | < δ ≤ . This proves that lim x → x f ( x ) = x- n ....
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This note was uploaded on 08/27/2011 for the course MATH 117 taught by Professor Akhmedov,a during the Fall '08 term at UCSB.

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HW 6 Solutions - Math 117 F 09 HW#6(sample solutions P.79#12 The symbol | f | denotes the function whose outputs are the absolute values of the

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