Complex differentiability is exactly a first-order complex-linear approximation with error $o(h)$.Suggest a slogan
Let \(f\) be defined on a neighborhood of \(z_0\in\CC\). Then \(f\) is holomorphic at \(z_0\) if and only if there exist \(a\in\CC\) and a function \(\psi\) defined for small \(h\neq0\) such that \[ f(z_0+h)-f(z_0)-ah=h\,\psi(h),\qquad \psi(h)\to0\text{ as }h\to0. \] In this case \(a=f'(z_0)\), and \(f\) is continuous at \(z_0\).
If \(f'(z_0)\) exists, put \(a\coloneqq f'(z_0)\) and \(\psi(h)\coloneqq\frac{f(z_0+h)-f(z_0)}{h}-a\), which tends to \(0\). Conversely, dividing the displayed identity by \(h\) gives \(\frac{f(z_0+h)-f(z_0)}{h}=a+\psi(h)\to a\), so \(f'(z_0)=a\). Finally \(f(z_0+h)-f(z_0)=h\big(a+\psi(h)\big)\to0\) as \(h\to0\).