A function $f(x)$ is differentiable on $x_{0} \in \mathbb{R}$ if
$f'(x_{0}) = \lim_{h \to 0} \frac{f(x_{0} +h) - f(x_{0})}{h}$ exists

$\text{Show } f(x) = x^{2} \text{ is differentiable for all } x \in \mathbb{R}$ \begin{aligned} f'(x) &= \lim_{h \to 0} \frac{f(x +h) - f(x)}{h}\\ f'(x) &= \lim_{h \to 0} \frac{x^2 + h^2 + 2xh - x^2 }{h} \\ f'(x) &= \lim_{h \to 0} \frac{h^2 + 2xh}{h}\\ f'(x) &= \lim_{h \to 0} h + 2x\\ f'(x) &= 2x \quad \text{ for all } x \in \mathbb{R} \end{aligned}

Holomorphic
A function $f(z)$ defined on some open neibourhood of a point $z_{0} \in \mathbb{C}$ is said to be holomorphic at $z_{0}$ if the complex derivative $f'(z_{0}) = \lim_{h \to 0 } \frac{f(z_{0} + h) - f(z_{0})}{h}$ exists. We said $f$ is holomorphic on an open set $\Omega$ if it is holomorphic at every $z_{0} \in \Omega$ and we said $f$ is holomorphic in a closed set $\mathbf{C}$ if it is holomorphic on some open set $\Omega$ containing $\mathbf{C}$. Functions are holomorphic on all of $\mathbb{C}$ are said to be $\mathit{entire}$

$\text{Show } f(z) = z^2 \text{ is holomorphic}$ \begin{aligned} f'(z) &= \lim_{h \to 0} \frac{(z+h)^2 - z^2}{h} \\ f'(z) &= \lim_{h \to 0} \frac{z^2 + 2hz + h^2 - z^2}{h}\\ f'(z) &= \lim_{h \to 0} \frac{2hz + h^2}{h} \nonumber \\ f'(z) &= \lim_{h \to 0} 2z + h\\ f'(z) &= 2z \end{aligned}