## Question

###### Define L((0,0)) = {f : (0,0) + R : f is bounded and for every c â‚¬ (0,0), lim f(z) exists in R} ITc Give an example of & function f â‚¬ L((0,0)) such that f is not continuous. Of course you have to prove all your assertions: We equip L((0,0)) with the norm |lfllo sup Ifk) cC(0,0) b. Prove that L((0,0) is a Banach space_ Hint: Use Theorem 7.9 and argue in a similar way as in Theorem 10.4_ You may use that lim f (z) exists ctc if and only if (f(Tn))nz1 is Cauchy for every (Tn)nz1 with Tn â‚¬

Define L((0,0)) = {f : (0,0) + R : f is bounded and for every c â‚¬ (0,0), lim f(z) exists in R} ITc Give an example of & function f â‚¬ L((0,0)) such that f is not continuous. Of course you have to prove all your assertions: We equip L((0,0)) with the norm |lfllo sup Ifk) cC(0,0) b. Prove that L((0,0) is a Banach space_ Hint: Use Theorem 7.9 and argue in a similar way as in Theorem 10.4_ You may use that lim f (z) exists ctc if and only if (f(Tn))nz1 is Cauchy for every (Tn)nz1 with Tn â‚¬ (0,c) and limn-o Tn

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