MATH 411 : MATH411_BOYLE-M_FALL2011_0101_MID_EXAM
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Math 411 fall 2011 boyle exam 1: (10 points) a metric space is a set x together with a real- valued function d de ned on pairs of points (x, y) from x. Prove that f has directional derivatives in every direction at (0, 0). (20 points) let h = {(x, y) r2 : y = 0}, the hori- Suppose f : r2 r2 is continuous and there are sequences pn and qn in h such that limn f (pn) = 0 and limn ||f (qn)|| = . Suppose also the partial derivative functions are bounded: there is a positive number m such that | f. Y (p)| m at every point p in r2. 7. (25 points) for each of the following, answer true or. A function f : rn r is continuously di erentiable if it a. is di erentiable and in addition it is continuous.