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hw1 - g to a b 6 Suppose f is a real Function on −∞ ∞...

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Homework 1 – Math 118B, Winter 2010 Due on Thursday, January 14, 2010 1. Let f be defined for all real x , and suppose that M > 0 and ǫ > 0 such that | f ( x ) f ( y ) | ≤ M | x y | 1+ ǫ , x, y R . Prove that f is constant. 2. Suppose that f ( x ) > 0 in ( a, b ). Prove that f is strictly increasing in ( a, b ), and let g be its inverse function. Prove that g is differentiable, and that g ( f ( x )) = 1 f ( x ) x ( a, b ) . 3. Suppose g is a real function on R , with bounded derivative (say | g | ≤ M ). Fix ǫ > 0, and define f ( x ) = x + ǫg ( x ). Prove that f is one-to-one if ǫ is small enough. 4. Suppose (a) f is continuous for x 0, (b) f ( x ) exists for x > 0, (c) f (0) = 0, (d) f is monotonically increasing. Put g ( x ) = f ( x ) x ( x > 0) , and prove that g is monotonically increasing. 5. Let g be defined in ( a, b ), with bounded derivative, i.e. | g ( x ) | ≤ M x ( a, b ) . (a) Prove that g is uniformly continuous in ( a, b ). (b) Prove that there exists a continuous extension of
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Unformatted text preview: g to [ a, b ]. 6. Suppose f is a real Function on ( −∞ , ∞ ). We say that x ∈ R is a fxed point For f iF f ( x ) = x . 1 2 (a) If f is diFerentiable and f ′ ( t ) n = 1 for every real t , prove that f has at most one ±xed point. (b) Show that the function f de±ned by f ( t ) = t + 1 1 + e t has no ±xed point, although 0 < f ′ ( t ) < 1 for all real t . (c) Prove that if there is a constant 0 < A < 1 such that | f ′ ( t ) | ≤ A for all t ∈ R , then f has a ±xed point x . To do this, given x 1 ∈ R arbitrary, construct the sequence x n +1 = f ( x n ) n ≥ 1 , and prove that the sequence converges to some point x . Then prove that x is the ±xed point....
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