Uniform continuity of entropy rate with respect to the $\bar{f}$-pseudometric

Assume that a sequence $x=x_0{x}_1\dots$ is frequency-typical for a finite-valued stationary stochastic process $\mathbf{\text{X}}$. We prove that the function associating to $x$ the entropy-rate $\bar{H}(\mathbf{\text{X}})$ of $\mathbf{\text{X}}$ is uniformly continuous when one endows the set of all frequency-typical sequences with the $\bar{f}$ pseudometric. As a consequence, we obtain the same result for the $\bar{d}$ pseudometric. We also give an alternative proof of the Abramov formula for the Kolmogorov-Sinai entropy of the induced measure-preserving transformation.