Exercise · ZW-COMPLEX-CONTINUITY-009 · ★★★ Advanced
If \(f\) is continuous on the closed disk \(|z|\le 1\), which claim is certain?
Choose one option.
Show answer
- C. \(|f|\) attains a maximum at some point of the disk
Explanation
The closed disk is closed and bounded, and \(|f|\) is continuous and real-valued: it attains its maximum (Weierstrass theorem). Continuity does not imply holomorphy.
Review the concept: Complex continuity
A function \(f\) is continuous at a point \(z_0\) of its domain if \(f(z)\) tends to the value \(f(z_0)\) as \(z\) tends to \(z_0\); it is continuous if it is so at every point of its domain.
\[\lim_{z\to z_0}f(z)=f(z_0)\]
More practice problems: complex continuity
- ZW-COMPLEX-CONTINUITY-001 At which points is f(z)=(1)/(z-2) continuous?
- ZW-COMPLEX-CONTINUITY-002 For f(z)=(1)/(z^2+1), which statement is correct?
- ZW-COMPLEX-CONTINUITY-003 Let f(z)=(z^2-4)/(z-2) for z≠ 2 and f(2)=c. Which c makes it continuous at 2?
- ZW-COMPLEX-CONTINUITY-004 The function f(z)=bar z is:
- ZW-COMPLEX-CONTINUITY-005 Let f(z)=(z^2)/(|z|) for z≠ 0 and f(0)=0. Is it continuous at 0?
- ZW-COMPLEX-CONTINUITY-006 Let f(z)=z^2 and g(w)=(1)/(w-4). At which points is g(f(z)) continuous?
- ZW-COMPLEX-CONTINUITY-007 Let f=u+iv with u=(xy)/(x^2+y^2), v=x^2 for z≠ 0 and f(0)=0. Is it continuous at 0?
- ZW-COMPLEX-CONTINUITY-008 Compute lim _z→ 1+ibig(Re(z^2)+Im(bar z)big).