Limits & Continuity Summary

Complex Analysis · Axiom Academy

SUMMARY Limits and Continuity Summary Section 4 review — complex limits, continuity, behavior at infinity, and the Riemann sphere. Complex limits require convergence from ALL directions — infinitely many paths, not just left and right. To disprove a limit exists, find two paths to z_0 that give different values. Continuity of f = u + iv reduces to continuity of the real components u and v . The extended complex plane has only ONE point at infinity. The Riemann sphere visualizes as a compact space — the distance to is finite. To analyze f at , study g(w) = f(1/w) as . A polynomial has a pole at ; e^ z has an essential singularity at . means that for every there is a with f(z) inside the disk of radius about L whenever z is inside the punctured disk of radius about z_0 . Path independence: if the limit exists it must be the same along every path to z_0 . Watch out for: two paths with different values prove the limit does NOT exist. All the standard limit laws — sum, product, quotient, and composition — carry over to complex limits exactly as in real analysis, provided the individual limits exist (and a denominator limit is nonzero). When to use: break a complicated limit into pieces you already know. f is continuous at z_0 when its limit there equals its value. In component form, f = u + iv is continuous iff both real functions u(x,y) and v(x,y) are continuous. Always continuous on their domains: polynomials, rationals (off poles), exponentials, and trig functions.

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