= Solution
\b[The <covariance> is zero.] Let $\mathcal F_t=\sigma(\varepsilon_s:s\leq t)$. The stationary noise-series representation makes $f(X_t)$ measurable with respect to $\mathcal F_t$, whereas the <martingale difference sequence> property gives $\mathbb E[X_{t+h}\mid\mathcal F_{t+h-1}]=0$. The <law of total expectation> therefore gives
$$
\mathbb E[X_{t+h}f(X_t)]
=\mathbb E\!\left[f(X_t)\mathbb E(X_{t+h}\mid\mathcal F_{t+h-1})\right]=0.
$$
Since $\mathbb EX_{t+h}=0$, this is the required <covariance>. All products are integrable by the <Cauchy-Schwarz inequality>, using finite <second moments> of $X$ and the stipulated square integrability of $f(X_t)$.
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