Fall 2014 - Problem 2

Solution.

Let $C \subseteq \br{0, 1}$ be a fat Cantor set of Lebesgue measure, say, $\frac{1}{2}$ and consider $f = \chi_C$ and let $x \in C$.

Because $C$ has no isolated points, we have a sequence $\set{y_n}_n \subseteq C$ such that $y_n \to x$. On the other hand, because $C$ is nowhere dense, we get a sequence $\set{z_n}_n \subseteq C^\comp$ such that $z_n \to x$ also. Thus, $f\p{y_n} \to 1$ and $f\p{z_n} \to 0$, so $x$ is not a point of continuity of $f$. This is true for any $x \in C$, and so $f\p{x + y}$ fails to converge to $f\p{x}$ on a set of positive measure.