Let $f$ be an integrable function on $\mathbb{R}$ where $\operatorname{support}(\widehat{f}) \subseteq [-\gamma, \gamma]$ for some $ 0 < \gamma < 1$
Prove that $\lvert f(x) - f(0)\rvert \leq c \gamma \lvert x\rvert \sup\limits_{ y \in \mathbb{R}}\left\{(1+|y|)\lvert f(y)\rvert\right\}$ for some absolute constant $c$.