Since, for every limit point x of E, there is a sequence {x

_{n}} in E such that limx

_{n}= x, I define g as below.

g(x)=f(x) if x in E; g(x)=lim

_{n}f(x

_{n}) if x belongs to (the closure of E)\E.

Would it be a right start? If so, then how do I prove that g is defined and continuous on (the closure of E)\E by using the fact that f is uniformly continuous?