Continuity in Two Dimensions

Complex Analysis · Axiom Academy

What "no breaks" means when the input and the output are both points in the plane. The whole idea, before any symbols On the real line, "continuous" has a friendly picture: you can draw the graph without lifting your pencil — no sudden jumps, no missing points. But a complex function takes a point in the plane to another point in the plane . There's no single curve to trace, so what could "no breaks" even mean here? Here's the honest version, and the only thing you ever need to remember: a function is continuous at a point z 0 when a small nudge to the input z only ever causes a small nudge to the output f(z) . Watch a point z wander in a tiny loop around z 0 on the left; its image w = z² is drawn at its true position on the right. The loop stays small, so the image stays small — it tracks along, never snapping away. Small in, small out — the image of a tiny loop is itself a tiny loop. That is continuity, before we ever write a limit. The catch: it has to work from every direction On the real line you only approach a point two ways — from the left and from the right. In the plane you can come in from infinitely many directions, and continuity demands the output settle to the same value no matter which one you pick. Drag to spin the approach direction. A point slides in along that ray toward z 0 , and its image w = z² is plotted. Notice the image always homes in on the one landing point f(z 0 ) — the gap |f(z) − f(z 0 )| shrinks to zero whichever way you came.

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