If f is defined and uniformly continuous on E, can somebody show me that there's a function g defined and continuous on the closure of E such that g = f on E?

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?