You don't just need the CR equations to hold, you need the partials to be continuous there! Here is question a. worked out:
$u(x,y)=yx$, $v(x,y)=y^2$. Thus, $\partial_xu=y\\ \partial_yu=x\\ \partial_xv=0\\ \partial_yv=2y $ The partials are continuous everywhere. The Cauchy Riemann equations give: $\partial_xu=\partial_yv\Leftrightarrow y=2y\Leftrightarrow y=0 \\ \partial_yu=-\partial_xv\Leftrightarrow x=0 $ $f$ is only complex differentiable at $(x,y)=0$ and $f^{\prime}(z)=0$
For b: $u(x,y)=x^2$, $v(x,y)=y^2$. Thus, $\partial_xu=2x\\ \partial_yu=0\\ \partial_xv=0\\ \partial_yv=2y $ The partials are continuous everywhere. The Cauchy Riemann equations give: $\partial_xu=\partial_yv\Leftrightarrow 2x=2y\Leftrightarrow x=y \\ \partial_yu=-\partial_xv\Leftrightarrow 0=0 $ $f$ is only complex differentiable at $S=\left\{(x,x)\in \mathbb{C}:x\in \mathbb{R}\right\}$