For the "not uniformly continuous" part: do you know how to formally negate a logical proposition, i.e. exchange "for-all's" and "there-exists's" and reverse (in)equalities? Try that with the definition of uniform continuity.
I actually discuss this in my post here. In essence, (we'll speak less generally now) if (0,1]\to\mathbb{R}" alt="f(0,1]\to\mathbb{R}" /> were uniformly continuous, then we could extend it to some uniformly continuous map . But, this is just nonsense since we'd have to have that