Addendum:

If we have a functionfcontinuous on [a,b] is it true thatfis uniformly continuous on (a,b) assumingf(a)andf(b)are defined? If so why?

What about iff(a) is not defined? For examplef(x) = 1/(x^2) is continuous on [0,1] (and so is uniformly continuous on [0,1]) but isfuniformly continuous on (0,1) consideringf(0) is not defined? If so why?