Proving Continuity of exp(z)

Complex Analysis · Axiom Academy

EXAMPLE Proving Continuity of e^z A complete – proof that the complex exponential is continuous everywhere. Show that the complex exponential is continuous at every point — that is, prove . (This makes an entire function: continuous everywhere.) Continuity means: for every target tolerance (right disk), we can find an input radius (left disk) so that every z within of z₀ lands within of e₀₀ . The proof below constructs that explicitly. Since z_0 was an arbitrary point of , e^z is continuous at every point — so is entire. Nice work — you built a full – proof that e^z is continuous, by factoring, bounding a power series, and choosing explicitly. Factor out the value: isolates the "small" part near z_0 . Modulus of the exponential: turns a complex factor into a real constant. Power-series bound: for , with . The construction: keeps and forces the final estimate below . The same factor-and-bound technique shows that , , and every entire function is continuous — and the bound being uniform on any bounded set means is in fact uniformly continuous there.

This is the written version of the interactive lesson above. See the full Complex Analysis course.