How to prove that this function satisfies the Cauchy Riemann equations? Let$$
f(z) = \left\{
        \begin{array}{ll}
            g(z) &amp; \quad z \neq 0 \\
            h(z) &amp; \quad z = 0
        \end{array}
    \right.
$$

where $g(z)=U_1+iV_1$ and $h(z)=U_2+iV_2$.

When we are asked to prove that $f(z)$ satisfies Cauchy Riemann equations at $z=0$ does that mean that we need to prove that ${U_1}_x={V_1}_y$ and ${U_1}_y=-{V_1}_x$ or ${U_2}_x={V_2}_y$ and ${U_2}_y=-{V_2}_x$?

If $h(z)=0$ does that mean that $f(z)$ satisfies Cauchy Riemann equations at $z=0$ for any $g(z)$?
The function $h$ is just a constant: it has little to do with the value of $f$, except, of course, that $f$ has to be continuous. All you need to do is check the Cauchy-Riemann equations for $g$ at $0$ (take the limit at $0$: it should exist if it is going to be analytic).
The function $h$ is just a constant: it has little to do with the value of $f$, except, of course, that $f$ has to be continuous. All you need to do is check the Cauchy-Riemann equations for $g$ at $0$ (take the limit at $0$: it should exist if it is going to be analytic).