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

Assume that a sequence $x=x_0x_1\ldots$ is frequency-typical for a finite-valued stationary stochastic process $\textbf X$. We prove that the function associating to $x$ the entropy-rate $\bar H(\textbf X)$ of $\textbf X$ 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.