TEAM PROJECT. Limit, Continuity, Derivative (a) Limit.

Chapter 13, Problem 13.3

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TEAM PROJECT. Limit, Continuity, Derivative (a) Limit. Prove that (I) is equivalent to the pair of relations lim Re i(z) = Re t, lim 1m Ie::) = 1m l. 2-----;"2'0 Z-Zo (b) Limit. If lim I(:::) exists, show that this limit is unique. z-zo (e) Continuity. If:::}o ::2' ... are complex numbers for which lim ::" = a, and if i(:) is continuous at 'it_CO z = a, show that lim i(::n) = i(a). n-----'""x (d) Continuity. If if:::) is differentiable at :::0' show that if:::) is continuous at :::0' (e) Differentiability. Show that if::) = Re z = x is not differentiable at any z. Can you find other such functions? (l) Differentiability. Show that if::) = 1:::12 is differentiable only at:: = 0; hence it is nowhere analytic.