  Erd\H{o}s \cite{MR168482} proved that the Continuum Hypothesis (CH) is
equivalent to the existence of an uncountable family $\mathcal{F}$ of (real or
complex) analytic functions, such that $\big\{ f(x) \ : \ f \in \mathcal{F}
\big\}$ is countable for every $x$. We strengthen Erd\H{o}s' result by proving
that CH is equivalent to the existence of what we call \emph{sparse analytic
systems} of functions. We use such systems to construct, assuming CH, an
equivalence relation $\sim$ on $\mathbb{R}$ such that any "analytic-anonymous"
attempt to predict the map $x \mapsto [x]_\sim$ must fail almost everywhere.
This provides a consistently negative answer to a question of Bajpai-Velleman
\cite{MR3552748}.
