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

f:C→C,z_0 ∈C such that f(z)−f(z_0 )=(z−z_0 )f(z−z_0 ) does lim_(z→z_0 ) f(z)=f(z_0 )?

$${f}:\mathbb{C}\rightarrow\mathbb{C},{z}_{\mathrm{0}} \in\mathbb{C}\:\mathrm{such}\:\mathrm{that} \\ $$$${f}\left({z}\right)−{f}\left({z}_{\mathrm{0}} \right)=\left({z}−{z}_{\mathrm{0}} \right){f}\left({z}−{z}_{\mathrm{0}} \right) \\ $$$$\mathrm{does}\:\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}\right)={f}\left({z}_{\mathrm{0}} \right)? \\ $$

Commented by 123456 last updated on 24/Jul/15

f′(z_0 )=^? f(0)

$${f}'\left({z}_{\mathrm{0}} \right)\overset{?} {=}{f}\left(\mathrm{0}\right) \\ $$

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

lim_(z→z_0 ) ((f(z)−f(z_0 ))/(z−z_0 ))=f ′(z_0 )=lim_(z→z_0 ) f(z−z_0 )=f(0)

$$\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}\:\frac{{f}\left({z}\right)−{f}\left({z}_{\mathrm{0}} \right)}{{z}−{z}_{\mathrm{0}} }={f}\:'\left({z}_{\mathrm{0}} \right)=\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}−{z}_{\mathrm{0}} \right)={f}\left(\mathrm{0}\right) \\ $$

Commented by 112358 last updated on 24/Jul/15

f(z)=(z−z_0 )f(z−z_0 )+f(z_0 ) ∴lim_(z→z_0 ) f(z)=lim_(z→z_0 ) [(z−z_0 )f(z−z_0 )+f(z_0 )] =lim_(z→z_0 ) (z−z_0 )lim_(z→z_0 ) f(z−z_0 )+lim_(z→z_0 ) f(z_0 ) Let z=a+bi,z_0 =c+di where a,b,c,d∈R ⇒z−z_0 =(a−c)+(b−d)i ∴lim_(z→z_0 ) (z−z_0 )=lim_(a→c) (a−c)+ilim_(b→d) (b−d)=0+0i=0∈C lim_(z→z_0 ) f(z−z_0 )=f(0)∈C lim_(z→z_0 ) f(z_0 )=f(z_0 )∈C (lim_(x→x_0 ) c=c) ∴lim_(z→z_0 ) f(z)=f(z_0 )∈C

$${f}\left({z}\right)=\left({z}−{z}_{\mathrm{0}} \right){f}\left({z}−{z}_{\mathrm{0}} \right)+{f}\left({z}_{\mathrm{0}} \right) \\ $$$$\therefore\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}\right)=\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}\left[\left({z}−{z}_{\mathrm{0}} \right){f}\left({z}−{z}_{\mathrm{0}} \right)+{f}\left({z}_{\mathrm{0}} \right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}\left({z}−{z}_{\mathrm{0}} \right)\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}−{z}_{\mathrm{0}} \right)+\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}_{\mathrm{0}} \right) \\ $$$${Let}\:{z}={a}+{bi},{z}_{\mathrm{0}} ={c}+{di}\:\: \\ $$$${where}\:{a},{b},{c},{d}\in{R} \\ $$$$\Rightarrow{z}−{z}_{\mathrm{0}} =\left({a}−{c}\right)+\left({b}−{d}\right){i} \\ $$$$\therefore\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}\left({z}−{z}_{\mathrm{0}} \right)=\underset{{a}\rightarrow{c}} {\mathrm{lim}}\left({a}−{c}\right)+{i}\underset{{b}\rightarrow{d}} {\mathrm{lim}}\left({b}−{d}\right)=\mathrm{0}+\mathrm{0}{i}=\mathrm{0}\in\mathbb{C} \\ $$$$\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}−{z}_{\mathrm{0}} \right)={f}\left(\mathrm{0}\right)\in\mathbb{C}\:\:\:\:\:\:\:\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}_{\mathrm{0}} \right)={f}\left({z}_{\mathrm{0}} \right)\in\mathbb{C}\:\:\:\left(\underset{{x}\rightarrow{x}_{\mathrm{0}} } {\mathrm{lim}}{c}={c}\right) \\ $$$$\therefore\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}\right)={f}\left({z}_{\mathrm{0}} \right)\in\mathbb{C} \\ $$$$ \\ $$

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