To check that $T_f$ is a distribution of zeroth order, we shall use condition (3) on this page. First, it is clear that $T_f$ is a linear mapping. To see that $T_f$ is continuous, suppose $K$ is a compact set in $U$ and $u\in \cD_K$ , i.e., $u$ is a smooth function with support in $K$ . We then have \begin{eqnarray*} |T_f(u)| &=& |\int_K f(x) u(x) dx |\\ &\le& \int_K |f(x)| \,\, |u(x)| dx \\ &\le& \int_K |f(x)| dx \, ||u||_\infty. \end{eqnarray*}Since $f$ is locally integrable, it follows that $C=\int_K |f(x)| dx$ is finite, so $$ |T_f(u)| \le C ||u||_\infty.$$ Thus $f$ is a distribution of zeroth order ([1], pp. 381).

References

1

S. Lang, Analysis II, Addison-Wesley Publishing Company Inc., 1969.