ℝ- and ℂ-Differentiability

Abstract

Let z₀ = x₀+iy₀ = (x₀,y₀) be a point in $\mathbb{C}$ and f a function defined on a neighbourhood of z₀ (e.g., on an open disk $\varDelta(z_{0},\ r)$ for some r > 0) with values in $\mathbb{C}$. Write $f(z) = \mathrm{Re}\ f(z)+i\mathrm{Im}\ f(z) = u(z)+iv(z) = u(x,y)+iv(x,y)$.