Differentiability at a point, definition

Differentiability at a point

Let $X$ be a subset of $\mathbb{R}$, and let $x_0\in X$ be an element of $X$ that is also a limit point of $X$. Let $f:X\to \mathbb{R}$ be a function. Then, if the following limit:

converges to some $L$ then we say that $f$ is differentiable at $x_0$ on $X$ with derivative $L$. We write: $f^{\prime }(x_0):=L$.

If $x_0$ is not an element of $X$ or is not a limit point of $X$, or the limit does not converge, we leave $f^{\prime }(x_0)$ undefined and we say that $f$ is not differentiable at $x_0$ on $X$.