1. The problem statement, all variables and given/known data "Show that Complex-differentiable functions are always Real-differetiable." I missed the lecture where the proof of this was given and I'm trying to figure it out. 2. Relevant equations Cauchy-Riemann, ux = vy, uy = -vx 3. The attempt at a solution Suppose a function f(z) = u + iv is C-differentiable, so it satisfies the Cauchy-Riemann equations everywhere. So ux = vy, uy = -vx at all points. A function f(x,y) is R-differentiable if f(x,y) = f(x0,y0) + f(x0, y0)(x-x0)(y-y0) + O(Δx,Δy) and O is such that O(Δx,Δy)/(||(Δx,Δy)||) -> 0 as ||(Δx,Δy)|| -> 0. Firstly, my notes are a bit sketchy so it's possible my definition of R-diff is wrong. I'm not really sure how to use the C-R equations with this definition either. It's not as if I can just substitute some stuff in to get the required result... Any hints/tips/suggestions? Edit: Oh wait, I just realised I can apply to definition to R-diff to both u and v, since they're both functions of x and y. Will report back with where this leads. Edit 2: That didn't really go anywhere. I realised I should be using the C-R equations to show that f satisfies the definition of R-diff. I'm kind of lost.