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

Question

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

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.

Answer 1

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 $f(x) = 1/|x - x_0|$.