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

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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 ) = x 2 g ( x ), where g ( x ) is given by a power series which is absolutely convergent for all
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