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.
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The explanation is attached below in two files. The files are identical in content, only differ in format. The first is in MS Word XP Format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
Check out also:
http://homepages.ius.edu/WCLANG/m413/hw11.pdf (it has another nice proof)
Hope this helps