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P8 Let U be an open subset of IR\" and consider a function f: U > R2 with coordi- nate functions f1: U > R and f2: U > IR. (i) Let a be a point in the closure of U (i.e., either ("i e U or ii is in the boundary of U) Suppose limgna f1(:f:') 2 b1 and liming f2(a:') 2 b2. Prove that liming f (55) 2 (b1, b2) directly from the denition of a limit: show that for every eneighborhood2 of 5 ='(b1, b2) 6 1R2 there is some 6-neighborhood of (i E R" so that f takes every point if 76 {1' that is in the intersection of U with N5 (ii) to a point in N45). The argument of (i) proves that if the limit of the coordinate functions f1 and f2 exists, then the limit of f = (f1, f2) also exists. (ii) Use (i) to prove that if f1 and f2 airs both continuous at a point (i E U, then f is continuous at ('1