Proving lim z² = z₀²

Complex Analysis · Axiom Academy

A complete ε-δ proof that the squaring map is continuous everywhere in the complex plane. Prove rigorously, straight from the ε-δ definition of a limit, that That is: given any , produce a so that forces . Our job is to discover how must depend on . No matter how tight the target tolerance (the green disk around z_0^2 ), we must find a small enough input radius (the orange disk around z_0 ) so that every z within of z_0 lands within of z_0^2 . The whole proof is the recipe for that . Nicely done — you built a full ε-δ proof by working backward from the goal to discover , then stating it forward. The squaring map is continuous at every . Factor, then split the modulus: isolates the small factor |z - z_0| we control. Auxiliary bound: pre-committing to forces , which caps the variable factor: . The min construction: satisfies both requirements at once. Result: with that , , so the limit holds. The same three moves — factor, bound the leftover factor with an auxiliary , then take a min — handle most polynomial continuity proofs in .

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