Show that if $g$ is differentiabl at $x_0$ we get $|\int_{-\pi}^{\pi}F_N'(t)[g(x_0-t)-g(x_0)]dt|\le C\int_{-\pi}^{\pi}|F_N'(t)||t|dt$.

Question

From the assumption that $g$ is differentiable at $x_0$ we get $$|\int_{-\pi}^{\pi}F_N'(t)[g(x_0-t)-g(x_0)]dt|\le C\int_{-\pi}^{\pi}|F_N'(t)||t|dt$$. This is a statement from the proof of the lemma below from Stein and Shakarchi's Fourier Analysis. The only other assumption needed to show this may be that $g$ is continuous, but I don't see how we can get that $[g(x_0-t)-g(x_0)]/t$ is $O(1)$ when it is only differentiable at a point. I can see that it is locally bounded but as in shown in the final part of the proof, we need this bound for the whole interval $[-\pi,\pi]$. How can we get this? I would greatly appreciate any help.