Continuity of Complex Functions

Complex Analysis · Axiom Academy

LESSON Continuity of Complex Functions One equation decides it — the limit must exist and land exactly on the function's value. A function f is continuous at z_0 when the value f(z) settles toward f(z_0) as z approaches z_0 from any direction. In one line: the limit exists and equals the function's value That single statement quietly bundles three requirements. The animation moves z around z_0 and checks each one as f(z) tracks toward f(z_0) . 2. Which Functions Pass — Everywhere You almost never check the limit by hand. A short list of functions is continuous on all of , and continuity is closed under the usual combinations — so anything built from these is continuous too. The animation lights each one up as its curve sweeps across — unbroken, no gaps, continuous for every input. If f and g are continuous at z_0 , so are and . If g is continuous at z_0 and f at g(z_0) , then is continuous at z_0 . is continuous on all of — it is a sum of a composition of entire functions with another entire function. No point needs checking individually. 3. Where Continuity Breaks: a Pole A rational function P(z)/Q(z) is continuous everywhere . At a pole — a zero of the denominator — the value is undefined and |f(z)| blows up, so requirement 2 fails and continuity breaks at that single point. The denominator vanishes at z = i and z = -i . Everywhere else this function is continuous; at those two points it is not.

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