An example of a measurable function $f$ such that both $f_{\pm}$ have infinite integral on each interval

Question

Is there a measurable function $f:\mathbb{R}\to\mathbb{R}$ such that, for each non-empty interval $I$, both Lebesgue integrals $$ \int_{I}f_{\pm}(x) dx=\infty, $$ where $f_{+}=\max(0,f)$ and $f_{-}=\max(0,-f)$ are the positive and the negative part of $f$, respectively?