there exists a continuous function $g$ on $\mathbb{R}$ and $\int_{\mathbb{R}}|f-g|<\epsilon$. Can we say that $g$ is uniform

Question

Let $f$ be integrable over $\mathbb{R}$, and let $\epsilon>0$. Then there exists a continuous function $g$ on $\mathbb{R}$ which vanishes outside a bounded set and $\int_{\mathbb{R}}|f-g|<\epsilon$. Can we say that $g$ is uniformly continuous? why or why not?