Does this condition imply that $f$ is locally Lipschitz or Lipschitz? Suppose there are constants $\delta > 0$ and $M < \infty$ such that for all $x \in \mathbb{R}$, $|f(x+t)−f(x)| \leq M|t|$ for all $t \in (−\delta,\delta)$. Then is $f$ locally Lipschitz or Lipschitz on $\mathbb{R}$? And if so, why?
Thank you.

$f : X \to Y$ is locally Lipschitz if and olny if $$\forall x \in X,\, \exists \delta > 0,\, \exists M > 0 \,\forall t,\, |t| < \delta \Rightarrow |f(x+t) - f(x)| \le M |t|$$ In the definition $\delta$ and $M$ could depend on $x$. In your question they do not depend on $x$ which is stronger. 
