Is it true that if $\displaystyle f: [0;2\pi] \rightarrow \mathbb{R}$ and $\displaystyle f$is Lipschitz continuous then exists $\displaystyle C>0$ that$\displaystyle f'(x) \leq C $ for every $\displaystyle x \in [0;2\pi] $?
Is it true that if $\displaystyle f: [0;2\pi] \rightarrow \mathbb{R}$ and $\displaystyle f$is Lipschitz continuous then exists $\displaystyle C>0$ that$\displaystyle f'(x) \leq C $ for every $\displaystyle x \in [0;2\pi] $?
Being Lipschitz continuous does not imply that the derivative even exists. For example $\displaystyle f(x) = |x-\pi|$ is Lipschitz continuous in $\displaystyle [0,2\pi],$ but it is not differentiable at $\displaystyle \pi.$