S(1): Let ε=1, and let any δ>0 be given.
S(2): Let n be an integer > max(1, 1/δ), and set x=1/n and y=1/(n+1).
S(3): Both x and y belong to (0,1), and |x-y| = 1/n(n+1) < 1/n < δ.
S(4): However, |f(x)-f(y)| = |n-(n+1)| = 1 = ε
S(5): This contradicts the definition of uniform continuity (i.e., this satisfies the negation of that definition), so f is not uniformly continuos on (0,1).
Also please use the definition of uniform continuity in Real Analysis. That definition is allowed to use to solve the problem.
Uniform continuity is investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.