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SelfAssessment1

# SelfAssessment1 - Self-Assessment 1 Math 104A Fall 2010...

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Self-Assessment 1 – Math 104A, Fall 2010 Answer the following questions without looking in the book. If you do not feel comfortable answering them, read the corresponding sections in the book, and then solve the problems, again without looking in the book. 1. A function f : [ a, b ] R is said to satisfy a Lipschitz condition with Lipschitz constant L on [ a, b ] if, for every x, y [ a, b ], we have | f ( x ) f ( y ) | ≤ L | x y | . (a) Show that if f satisfies a Lipschitz condition with Lipschitz con- stant L on [ a, b ], then f is continuous on [ a, b ]. (b) Show that if f has a derivative that is bounded on [ a, b ] by L , then f satisfies a Lipschitz condition with Lipschitz constant L on [ a, b ]. (c) Give an example of a function that is continuous on a closed in- terval but does not satisfy a Lipschitz condition on the interval. 2. Let f C ([ a, b ]), and consider p ( a, b ). (a) Suppose f ( p ) negationslash = 0. Show that there exists a δ > 0 such that f ( x ) negationslash = 0 for all x [ p δ, p + δ ] [ a, b ].

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SelfAssessment1 - Self-Assessment 1 Math 104A Fall 2010...

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