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

Invariant linear functionals on $L^{\infty}(\mathbb{R}_+)$