GATE 2022 PH – Question 25
For real a, $f(z)=z+|z-a|^2$ is
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Correct answer: (A) continuous at (a,a); (C) complex-differentiable at (a,0)
Explanation
Write $f(z)=z+|z-a|^2$ with $z=x+iy$ and real $a$:
$$f=\left[x+(x-a)^2+y^2\right]+i\,y,\qquad u=x+(x-a)^2+y^2,\quad v=y .$$
**Continuity.** $u$ and $v$ are polynomials, so $f$ is continuous everywhere, including at $(a,a)$. ✓ (A)
**Cauchy-Riemann conditions** for complex differentiability:
- $u_x=1+2(x-a)$ and $v_y=1$, so $u_x=v_y$ requires $x=a$.
- $u_y=2y$ and $-v_x=0$, so $u_y=-v_x$ requires $y=0$.
Both hold only at the single point $(a,0)$. So:
- the function is complex-differentiable at $(a,0)$ ✓ (C),
- it is not differentiable at $(a,a)$ ✗ (B),
- it is differentiable only at one isolated point, so it is not analytic (analytic requires differentiability in a neighbourhood) ✗ (D).
Answer **A and C**.