If A is bounded and not compact, prove thrrr is a continuous function on A that is not uniformly continuous.

Question

I found the same question, but I am not understanding the function definition of why it's not u.c or how the not compactness criterion is used.

The exact quote that is on that page is "Prove that there is a function that is continuous on $A$ but not uniformly continuous." That does not mean that every continuous function is non-uniformly continuous. It just means that a continuous function is not necessarily uniformly continuous, but it could be. — Arthur, Mar 22 '17 at 22:38
$\exists\ne\forall$. Edit the question to correct the statement. — Martín-Blas Pérez Pinilla, Mar 22 '17 at 22:43

score 1 · Answer 1 · answered Mar 22 '17 at 22:48

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Hint: let be $x_0\not\in A$ but limit of a sequence of points of $A$ (alternatively, $x_0\in\partial A\setminus A$). Consider the function $f(x) = 1/|x - x_0|$.

answered Mar 22 '17 at 22:48

Martín-Blas Pérez Pinilla

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My problem is showing that f is not uniformly continuous. – user338365 Mar 22 '17 at 22:51
@user338365, idea: uniformly continuous functions preserve Cauchy sequences (http://math.stackexchange.com/questions/252538/how-do-i-prove-a-uniformly-continuous-function-preserves-cauchy-sequences). Construct a Cauchy sequence $(x_n)$ s.t. $(f(x_n))$ isn't a Cauchy sequence. – Martín-Blas Pérez Pinilla Mar 22 '17 at 22:55
ok I see now that contracting a Cauchy sequence in A with the limit point not being in A. – user338365 Mar 22 '17 at 23:05
but I am having trouble showing f(Xn) is not Cauchy. – user338365 Mar 22 '17 at 23:05
@user338365, is unbounded. – Martín-Blas Pérez Pinilla Mar 22 '17 at 23:50

If A is bounded and not compact, prove thrrr is a continuous function on A that is not uniformly continuous.

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