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mid2_6337 - Write your name clearly on all sheets of paper...

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Write your name clearly on all sheets of paper you will turn in and possibly number them. Write clearly and large enough to be easily readable. Your proofs must be complete and clearly written. All scalars appearing in the exercises are supposed to be real if not differently stated. Each question is worth the number of points indicated. Fifty points will grant you an A. 1 Let f ( x ) be a bounded measurable function from R n to R such that lim sup | f ( x ) ||k x k α = C < where k x k = p x 2 1 + · · · + x 2 n . a) (7 pts) For which values of α can you say that f L 1 ( R n ). b) (5 pts) Is your condition on α necessary? 2 Given f L 1 ( R ) define F ( x ) = R x -∞ f ( y ) dy . a) (7 pts) Show that F ( x ) is a continuous function. Is F ( x ) differentiable? b) (10 pts) Show that for every g C 1 0 ( R ) we have: Z R F ( x ) g 0 ( x ) dx = - Z R f ( x ) g ( x ) dx. Remember that C 1 0 ( R ) is the set of all function in C 1 ( R ) that are 0 outside a compact set. Moreover C 1 ( R ) is the set of all function that are continuous and admit a continuous derivative.
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