From the authors' introduction: ``We consider the question as to whether or not the hypothesis that $\lim_{t\rightarrow\infty}(f(t+s)-f(t))=\lambda s$ for all $s$ implies the conclusion that $\lim_{t\rightarrow\infty}f(t)/t=\lambda$ in the presence of appropriate auxiliary hypotheses. We show that the result is not valid without some additional hypothesis, and that either measurability of $f$ or boundedness of $f$ on an appropriate set of intervals is adequate.''