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(10 pts) Let E C R be a bounded set of reals. and f : E IR a uniformly continuous function. Prove that f is bounded. Evaluate the integral So x+y(y - 2x)² dy)) dar using change of variables u = x + y. v = y - 2.x. Define f : R² R by 1 - cosey) x2 + y2 if y = 0 f(x,y) = otherwise. (a) Show that f is continuous at (0,0). (b) Calculate all the directional derivatives of f at (0,0). (c) State the definition of differentiability for a function f : R² R. (d) Show that f is not differentiable at (0,0). Hint: violate the defi- nition.

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