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.© BrainMass Inc. brainmass.com March 4, 2021, 6:13 pm ad1c9bdddf
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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.