A necessary condition for principal cluster sets to be void

Let $f$ be an arbitrary function from the open unit disk $D$ into the Riemann sphere $W$, and let $p$ be a point on the unit circle $C$. We prove that if the principal cluster set of $f$ at $p$ is void, then either $p$ is an ambiguous point of $f$ or the diameter of each arc-cluster set of $f$ at $p$ is greater than a fixed positive number.