Unformatted text preview: f : S 7→ R such that f does not have a maximum value on S . 5. Suppose that f : R 7→ R and g : R 7→ R are diﬀerentiable functions. Prove that the diﬀerence fg is diﬀerentiable. Bonus. Prove that the product f · g is diﬀerentiable. 6. a. Suppose that f : ( a, b ) 7→ R is a decreasing function. Prove that for every c ∈ ( a, b ) either f is not diﬀerentiable at c , or f ( c ) ≤ 0. b. Suppose that f : ( a, b ) 7→ R is a diﬀerentiable function and and for all c ∈ ( a, b ), f ( c ) < 0. Prove that f is decreasing. Bonus. Give an example of a continuous function f : R 7→ R that is diﬀerentiable everywhere, and such that f (0) < 0, but f is not decreasing on any open interval containing 0. Prove that your example has indeed all the required properties....
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 Spring '07
 thieme
 Calculus, Topology, Continuous function, Metric space, uniformly continuous function

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