hw7 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 7 D UE...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 7 DUE: Thurs. Nov. 4, 2010. Late papers will be accepted until 1:00 PM Friday . Note: We say a function is smooth if its derivatives of all orders exist and are continuous. 1. Let f : [ a , ) R be a smooth function whose first derivative is bounded: | f ± ( x ) | ≤ M for all x a . Prove that it is uniformly continuous on [ a , ) . As immediate examples, x 1 / 3 is uniformly continuous for all x 1 and cos x is uniformly continuous for all x . 2. a) Show that sin x is not a polynomial. b) Show that sin x is not a rational function, that is, it cannot be the quotient of two polyno- mials. c) Let f ( t ) be periodic with period 1, so f ( t + 1 ) = f ( t ) for all real t . If f is not a con- stant, show that it cannot be a rational function. that is, f cannot be the quotient of two polynomials. d) Show that e x is not a rational function. 3. a) If a smooth function
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hw7 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 7 D UE...

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