D is a nonempty bounded subset of R. Let a=inf D and b=sup D.

Let f:D-->R be a uniformly continuous function.

Let g(x)={f(x) if x in D; if x=a; if x=b

Prove that exist such that g(x) is continuous.

Well, g(x) is continuous on (a,b) because f(x) is continuous, by hypothesis.

Can I take the limit as x-->a of f from the right to find and the limit as x-->b of f from the left to find , thus making g(x) continuous?