Question

Which of the following is true about f(z)=z^2+2z?

(a) Continuous and differentiable

(b) Continuous but not differentiable

(c) Neither continuous nor differentiable

(d) Differentiable but not continuous

The question was posed to me by my school teacher while I was bunking the class.

My doubt stems from Differentiability topic in division Complex Function Theory of Engineering Mathematics

Answer 1

The correct answer is (a) Continuous and differentiable

Best explanation: z=x+iy

In general the limits are discussed at origin, if nothing is specified.

f(x, y)=(x+iy)^2+2(x+iy)

f(x, y)=x^2-y^2+2xiy+2x+2iy

Left limit: \(\lim_{x \to 0 \\ y \to 0}\)x^2-y^2+2xiy+2x+2iy

\(\lim_{y \to 0}\)-y^2+2iy

=0

Right limit: \(\lim_{y \to 0 \\ x \to 0}\)x^2-y^2+2xiy+2x+2iy

\(\lim_{x \to 0}\)x^2+2x

=0

Both the limits are equal, therefore the function is continuous. To check differentiability,

f’(z)=\(\lim_{\delta z \to 0}\frac{f(z+\delta z)-f(z)}{\delta z}\) should exist.

\(\lim_{\delta z \to 0}\frac{(z+\delta z)^2+2(z+\delta z)-(z^2+2z)}{\delta z}\)

\(\lim_{\delta z \to 0}\frac{z^2+(\delta z)^2+2z(\delta z)+2z+2(\delta z)-z^2-2z}{\delta z}\)

=2z+2

Since f’(z) exists, the function is differentiable as well.