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Unformatted text preview: MATH 104, SUMMER 2006, HOMEWORK 6 SOLUTION BENJAMIN JOHNSON Due July 24 Assignment: Section 17: 17.6, 17.9(a), 17.10(d), 17.14 Section 18: 18.2, 18.6, 18.10 Section 17 17.6 Prove that every rational function is continuous. Proof. By exercise 17 . 5( b ), every polynomial function is continuous on R . By Theorem 17.4(iii), if f and g are continuous functions (on their domains) then f / g is continuous on dom( f ) ∩ { x ∈ dom( g ) : g ( x ) , } . So if p and q are polynomial functions, then p q is continuous on { x ∈ R : q ( x ) , } , (which is the domain of p q ). 17.9 Prove that each of the following functions is continuous at x using the δ property of continuity. (a) f ( x ) = x 2 , x = 2 Proof. Let > 0. Choose δ = min { 1 , 5 } . Let x ∈ R . Assume  x 2  < δ . Then since  x 2  < 1, we also have  x + 2  < 5, so...  x 2 2 2  =  x 2 4  =  x 2  x + 2  < 5 · 5 = 17.10 Prove that the following functions are discontinuous at the indicated points. (d) P ( x ) = 15 for 0 ≤ x < 1 and P ( x ) = 15 + 13 n for n ≤ x < n + 1, x a positive integer....
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 Spring '09
 Math, Continuous function, Metric space, Xn, benjamin johnson

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