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.
Hello and thank you for posting your question to Brainmass!
The explanation is attached below in two files. The files are identical in content, only ...
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.