Exercise · ZW-COMPLEX-CONTINUITY-006 · ★★☆ Intermediate
Let \(f(z)=z^2\) and \(g(w)=\dfrac{1}{w-4}\). At which points is \(g(f(z))\) continuous?
Choose one option.
Show answer
- B. On \(\mathbb{C}\setminus\{2,-2\}\)
Explanation
The composition \(1/(z^2-4)\) is continuous where \(f(z)\) lies in the domain of \(g\), i.e. \(z^2\neq 4\), so \(z\neq\pm 2\).
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-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).
- ZW-COMPLEX-CONTINUITY-009 If f is continuous on the closed disk |z|≤ 1, which claim is certain?