Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function and $0 < M < \infty$, prove that $f$ is Lipschitz with constant $M$, if for each $x \in \mathbb{R}$ we have $Df(x) \subset [-M,M]$.
Given a function $f : \mathbb{R} \rightarrow \mathbb{R}$, an extended real number $\lambda$ is called a derived number for $f$ at $x_0$ if there exists a sequence $h_n \rightarrow 0 (h_n \neq 0)$ such that $$\lim_{n \to \infty} \frac{f(x+h_n) - f(x)}{h_n} = \lambda \text{ or } \lambda = Df(x_0).$$
$Df(x_0)$ can have infinite values - consider the example $f(x) = |x|$, where $Df(0) = [-1,1]$
I think mean value theorem cannot be applied here, because differentiability is not guaranteed. Any help with this problem is appreciated.