Let $f=u+iv$ be analytic inside and on a simple closed contour $\gamma$. Show by an example that Cauchy's Theorem does not hold separately for the real and imaginary parts of $f$.
Honestly I have nothing to show as prelim here. Any comment would be much appreciated.
Let $\gamma$ be the square with corners $0,1,1+i,i$. Then $$ \int_{\gamma} z \, dz = 0, $$ but $$ \int_{\gamma} \Re(z) \, dz = \int_0^1 x \, dx + \int_1^{1+i} 1 \, dz + \int_{1+i}^i x \, dx + \int_{i}^0 0 \, dz = i, \\ \int_{\gamma} i\Im(z) = i\int_0^1 0 \, dx + i\int_1^{1+i} \Im(z) \, dz + i\int_{1+i}^i 1 \, dz + i\int_{i}^0 \Im(z) \, dz = -i $$