f differentiable in the complex setting means that f is holomorphic and hence that it satisfies the Cauchy-Riemann equations. Differentiating $u^2-v^2=c$ with respect to $x$ and to $y$ we obtain $uu_x-vv_x=0$ and $uu_y-vu_x=0$. Exploiting Cauchy-Riemann we get $uu_x+vu_y=0$ and $uu_y-vu_x=0$, which can be seen as a $2\times 2$ system in the unknowns $u_x$ and $u_y$. If the system has a unique solution we have that the determinant $u^2+v^2>0$, and $(u_x,u_y)=(0,0)$ is a solution, hence the unique solution. In this way we get u is constant and similarly that v is constant, imlying that f is constant. Otherwise the determinant is zero obtaining $u^2+v^2=0$, implying $u=v=0$, hence also f is identically zero. In both cases f is constant.