  The aim of this paper is to study the Frechet-Urysohn property of the space
$Q_p(X,\mathbb{R})$ of real-valued quasicontinuous functions, defined on a
Hausdorff space $X$, endowed with the pointwise convergence topology. It is
proved that under Suslin's Hypothesis, for an open Whyburn space $X$, the space
$Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable. In
particular, it is true in the class of first-countable regular spaces $X$. In
ZFC, it is proved that for a metrizable space $X$, the space
$Q_p(X,\mathbb{R})$ is Frechet-Urysohn if and only if $X$ is countable.
