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.

$f$ is Lipschitz.
To prove this, consider arbitrary $x,y \in \Bbb R$ with $x<y$. Divide the interval $[x,y]$ into $N$ small subintervals $[x_i,x_{i+1}]$ (with $x_{i+1}-x_i < \delta$, $x_0=x$ and $x_N=y$).
Then
$$|f(y)-f(x)| = |f(x_N)-f(x_0)| \le \sum_{i=0}^{N-1} |f(x_{i+1})-f(x_i)| \le M \sum_{i=0}^{N-1} |x_{i+1}-x_i| = M|y-x|$$
