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# GATE 2016 EE - SET 2 - Question 6

Let $z=x+iy$ and $z^{\ast }=x-iy$

$f(z)=x+iy+x-iy=2x$

A function is said to be continuous at a point x=a when the following conditions are satisfied.

(1) f(a) exists.

(2) $\lim_{x\to a}f(x)$ exists

(3) $f(a)=\lim_{x\to a}f(x)$

Therefore the function is continuous.

Let $f(z)=2z+i0 = u+iv$; $u = 2x, v = 0$

$\frac{\partial u}{\partial x}=2; \frac{\partial u}{\partial y}=0; \frac{\partial v}{\partial x}=0; \frac{\partial v}{\partial y}=0$

$\frac{\partial u}{\partial x} \neq \frac{\partial v}{\partial y}$

Therefore, the real and imaginary parts of f(z) do not satisfy Cauchy Riemann equations.

This implies that the function is not analytic.

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