  We consider a continuous version of the classical notion of Banach limits,
namely, positive linear functionals on $L^{\infty}(\mathbb{R}_+)$ invariant
under translations $f(x) \mapsto f(x+s)$ of $L^{\infty}(\mathbb{R}_+)$ for
every $s \ge 0$. We give its characterization in terms of the invariance under
the operation of a certain linear mapping on $L^{\infty}(\mathbb{R}_+)$.
Applications to summability methods are provided in the last section.
