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Problem-set (1) Suppose DCR, a is a limit point of D, f :D - R. and f(x) 0. Yr E D. If Iim f(x) = L, where L 0. prove that Iim Vf(r) = VL. Hint: Modify the proof of Proposition 2.1.9. (2) If n Z+ and f R - R is defined by f(x) = =*, prove that lim f(x) = a". Hint: Apply Proposition 5.1.4 and Let e > o be given. Choose & = . 0. Ire and I # a, them Uf(x) - a| = al i.e. |r-a|For problemas (4) and (5), you may use the following fact: (1) (4) Prove that = (a x # a if and oaly if 0 (5) Suppose DCR, a D. and f D - R. If f is continuos at a and f(a) > 0, show that there is an open interval I such that a €1 and f(x) >0, VI €/\D. 1

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