Prove that f is analytic at z_(0) if and only if f is analytic in some neighborhood of z_(0) .

Prove that f is analytic at z_(0) if and only if f is analytic in some neighborhood of z_(0) .
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Assume z₀=x₀+iy₀ (complex number).

Also, let y₀=f(x₀), so z₀=x₀+if(x₀). Limit of f(x) as x→x₀ is y₀ if f is analytic at z₀.

But this limit implies: 

Lim x→x₀⁻=Lim x→x₀⁺=y₀, therefore f is analytic in the neighbourhood of z₀.

by Top Rated User (1.2m points)

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