  In this paper, we prove that if $E$ is a uniquely remotal subset of a real
normed linear space $X$ such that $E$ has a Chebyshev center $c \in X$ and the
farthest point map $F:X\rightarrow E$ restricted to $[c,F(c)]$ is partially
statistically continuous at $c$, then $E$ is a singleton. We obtain a necessary
condition on uniquely remotal subsets of uniformly rotund Banach spaces to be a
singleton. Moreover, we show that there exists a remotal set $M$ having a
Chebyshev center $c$ such that the farthest point map $F:\mathbb{R}\rightarrow
M$ is not continuous at $c$ but is partially statistically continuous there in
the multivalued sense.
