Question

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

(a) Continuous and differentiable

(b) Continuous but not differentiable

(c) Neither continuous nor differentiable

(d) Differentiable but not continuous

I have been asked this question in an online interview.

This intriguing question comes from Differentiability in chapter Complex Function Theory of Engineering Mathematics

Answer 1

Correct option is (a) Continuous and differentiable

To explain: z=x+iy

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

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

f(x, y)=(x-y)+i(x+y)

Left limit: \(\lim_{x \to 0 \\ y \to 0}\)(x-y)+i(x+y)

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

=0

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

 \(\lim_{x \to 0}\)x+ix

= 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.

\(\lim_{\delta z \to 0}\frac{z+\delta z+iz+i\delta z-z+iz}{\delta z}\)

\(\lim_{\delta z \to 0}\frac{\delta z+i\delta z}{\delta z}\)

=z+i

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