We are supposed to decide whether a set of real functions $u_n(x+iy)$ can be extended to complex differentiable functions on $\mathbb{C}$ and give the extension.
My approach was to use the Cauchy-Riemann Equations and then integrate to find a real-valued $v_n$ such that $f=u_n(x+iy)+iv(x+iy)$ satisfies C-R.
My understanding of what "extending" means is a bit shaky. The function $f$ found using this method is by design holomorphic as long as $u$ and $v$ are differentiable. But in one example $f(1)=3+i\neq3=u(1)$
Is this still correct, even if $f(\mathbb{R})\neq u(\mathbb{R})$? Did I "extend" u properly or is this a fatal error?