Let $(E, \mathcal{A}, \mu)$ be a measure space, and let $f: E \longrightarrow \mathbb{R}$ be an integrable function.
Suppose that $(E, \mathcal{A}, \mu)=(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda)$, where $\lambda$ is Lebesgue measure. Show that the function $F$ defined by $$ F(x):= \begin{cases}\int_{[0, x]} f \mathrm{~d} \lambda & \text { if } x \geq 0, \\ -\int_{[x, 0]} f \mathrm{~d} \lambda & \text { if } x<0,\end{cases} $$ is uniformly continuous on $\mathbb{R}$. Can someone give me idea on how I can approach, thank you.