Let $f$ be an integrable function on $\mathbb{R}$ where support($\hat{f}$) $\subseteq$ [$-\gamma, \gamma$] for some $ 0 < \gamma < 1$

Prove that | $f(x) - f(0)$| $ \leq c \gamma$ |x| $\underset{ y \in \mathbb{R}}{sup}(1+|y|)|f(y)|$ for some absolute constant $c$.