I am reading this paper on Atsuji metric space. A metric space $X$ is said to be Atsuji if every real-valued continuous function on $X$ is uniformly continuous.
In theorem 3.7 , it is proved that [see 3.7(e)], if $X$ is an Atsuji space then for any real-valued continuous function $f$ on $X$, there is some natural number $n$ such that every point of $A=\{x:|f(x)|\ge n\}$ is isolated and $\inf\{I(x):x\in A\}>0.$
Now $[0,1]$ is an Atsuji space (being compact) and $f:[0,1]\to\mathbb R:x\mapsto0.5$ is continuous. But I cannot find any $n$ such that $A=\{x:|f(x)|\ge n\}$ is isolated. How is it possible that there is no such $n$ that would make $A$ isolated?