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Show that the two definitions of continuity in section 2.1 are equivalent. Consider separately the cases where...

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Advance Math

Show that the two definitions of continuity in section 2.1 areequivalent. Consider separately

the cases where z0 is an accumulation point of G and where z0 isan isolated point of G.

2.1 :

Definition1. Suppose f : G → C. If z0 ∈ G and either z0 is anisolated point of G or lim f(z) = f(z0) (z→z0)
then f is continuous at z0. More generally, f is continuous on E ⊆G if f is continuous at every z ∈ E.

Definition 2.

Suppose f : G → C and z0 ∈ G. Then f is continuous at z0 if, forevery positive real

number ε there is a positive real number δ so that
|f(z)−f(z0)|<ε for all z∈G satisfying |z−z0|<δ.

Thanks.

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