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Question Number 1339 by 123456 last updated on 24/Jul/15

f : C \to C, z_{0} \in C such that

f (z) - f (z_{0}) = (z - z_{0}) f (z - z_{0})

does \lim_{z \to z_{0}} f (z) = f (z_{0}) ?

Commented by 123456 last updated on 24/Jul/15

f^{'} (z_{0}) \overset{?}{=} f (0)

Commented by prakash jain last updated on 24/Jul/15

\lim_{z \to z_{0}} \frac{f (z) - f (z_{0})}{z - z_{0}} = f^{'} (z_{0}) = \lim_{z \to z_{0}} f (z - z_{0}) = f (0)

Commented by 112358 last updated on 24/Jul/15

f (z) = (z - z_{0}) f (z - z_{0}) + f (z_{0})

∴ \lim_{z \to z_{0}} f (z) = \lim_{z \to z_{0}} [(z - z_{0}) f (z - z_{0}) + f (z_{0})]

= \lim_{z \to z_{0}} (z - z_{0}) \lim_{z \to z_{0}} f (z - z_{0}) + \lim_{z \to z_{0}} f (z_{0})

L e t z = a + b i, z_{0} = c + d i

w h e r e a, b, c, d \in R

\Rightarrow z - z_{0} = (a - c) + (b - d) i

∴ \lim_{z \to z_{0}} (z - z_{0}) = \lim_{a \to c} (a - c) + i \lim_{b \to d} (b - d) = 0 + 0 i = 0 \in C

\lim_{z \to z_{0}} f (z - z_{0}) = f (0) \in C \lim_{z \to z_{0}} f (z_{0}) = f (z_{0}) \in C (\lim_{x \to x_{0}} c = c)

∴ \lim_{z \to z_{0}} f (z) = f (z_{0}) \in C

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