While studying differential equations, I came across the existence and uniqueness of solutions for a differential equation. Then after studying some theorems these statements are troubling me.
Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function. If $f$ is Lipschitz continuous, does that mean that $f'$ is bounded?
What if $f$ is defined on a compact subset of $\mathbb{R}$?
Yes, it does. For all $x\in\mathbb R$ you have $$ f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ and the term in the limit is bounded (uniformly in $x$), since by Lipschitz continuity you have $$ |f(x+h)-f(x)|\le C |(x+h)-x|=C |h|$$ for all $x\in\mathbb R$, where $C$ is a Lipschitz constant for $f$.
It is completely irrelevant, whether $f$ is defined on the entire space $\mathbb R$ or on any subset.