TEAM PROJECT. Limit, Continuity, Derivative (a) Limit. Prove that (I) is equivalent to

Chapter 13, Problem 13.1.81

(choose chapter or problem)

Limit, Continuity, Derivative

(a) Limit. Prove that (1) is equivalent to the pair of relations

\(\lim _{z \rightarrow z_{0}} \operatorname{Re} f(z)=\operatorname{Re} l, \quad \lim _{z \rightarrow z_{0}} \operatorname{Im} f(z)=\operatorname{Im} l\).

(b) Limit. If \(\lim _{z \rightarrow z_{0}} f(z)\) exists, show that this limit is unique.

(c) Continuity. If \(z_{1}, z_{2}, \cdots\) are complex numbers for which \(\lim _{n \rightarrow \infty} z_{n}=a\), and if f(z) is continuous at z = a, show that \(\lim _{n \rightarrow \infty} f\left(z_{n}\right)=f(a)\).

(d) Continuity. If f(z) is differentiable at z = - show that f(z) is continuous at \(z_{0}\).

(e) Differentiability. Show that f(z) = Re z = x is not differentiable at any z. Can you find other such functions?

(f) Differentiability. Show that \(f(z)=|z|^{2}\) is differentiable only at z = 0; hence it is nowhere analytic.

Text Transcription:

lim _{z rightarrow z_0} Re f(z) = Re l,      lim _{z rightarrow z_0} Im f(z) = Im l

lim _{z rightarrow z_0} f(z)

z_1, z_2, cdots

lim _n rightarrow infty z_n = a

z_0

lim _n rightarrow infty f(z_n) = f(a)

f(z) = | z|^2