Question

Which of the following is true about f(z)=\(\frac{z+iz}{z^2}\)?

(a) Continuous and differentiable

(b) Continuous but not differentiable

(c) Neither continuous nor differentiable

(d) Differentiable but not continuous

I had been asked this question by my school principal while I was bunking the class.

The origin of the question is Differentiability in section Complex Function Theory of Engineering Mathematics

Answer 1

The correct choice is (c) Neither continuous nor differentiable

The explanation: z=x+iy

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

f(x, y)=\(\frac{x+iy+ix+i^2y}{(x+iy)^2}\)

f(x, y)=\(\frac{i(x+y)+(x-y)}{(x+iy)^2}\)

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

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

=does not exist

Since, the left limit itself does not exist, the function is not continuous. If a function is not continuous, it cannot be differentiable as well.