sheet3 - CM221A ANALYSIS I Exercise Sheet 3 1. Give a...

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Unformatted text preview: CM221A ANALYSIS I Exercise Sheet 3 1. Give a complete proof from the definition of continuity that a polynomial is a continuous function on R. (You are supposed to give a rigorous proof, using ε and δ .) √ 2. Show that f (x) = x is a continuous function on the interval [0, +∞). √ √ √ √ Hint: use the identity x − y = (x − y )( x + y )−1 . 3. Give an example of a nonnegative function f : R → R which is continuous at every x ∈ R but does not attain a minimum value. 4. Write down the power series for the function f (x) = (sin x) − x. Prove that f (x) = x2 g (x), where g (x) is given by a power series which is absolutely convergent for all x ∈ R. 5. Using Q5, prove that lim sin x = 1. x→0 x Hint: prove that |g (x)| ≤ C for all x ∈ [−1, 1] where C is some positive constant, and deduce the required result from the estimate | sin x − x| ≤ C x2 . 6. Sketch the graph of the function f (x) = x sin(1/x). Is this function continuous at x = 0? Hint: use Q5 to find out what happens as x → ∞. 7. Use the result of Q5 to find all values of the positive integers m, n for which the function f (x) = x−m sin(xn ) is bounded on (0, 1). 1 ...
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This document was uploaded on 01/31/2011.

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