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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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