Derivative at an Interior Point

Let $I$ be an interval, let $f:I\to\mathbb{R}$, and let $x_0\in I$ be an interior point of $I$. The function $f$ is differentiable at $x_0$ if there exists a real number $L$ such that for every $\varepsilon>0$ there exists $\delta>0$ with the following property: whenever $h\in\mathbb{R}$ satisfies $0<|h|<\delta$ and $x_0+h\in I$, one has $\left|\frac{f(x_0+h)-f(x_0)}{h}-L\right|<\varepsilon.$ In that case $L$ is called the derivative of $f$ at $x_0$ and is denoted by $f'(x_0)$.