Differentiable function/D in R/Via limit/Definition

Differentiability

Let ${}D\subseteq \mathbb {R}$ be a subset, ${}a\in D$ a point, and

$f\colon D\longrightarrow \mathbb {R}$

a function. We say that ${}f$ is differentiable in ${}a$ if the limit

$\operatorname {lim} _{x\in D\setminus \{a\},\,x\rightarrow a}\,{\frac {f(x)-f(a)}{x-a}}$

exists. In the case of existence, this limit is called the derivative of ${}f$ in ${}a$, written

$f'(a).$