Let g: [-1,1] --> R be a continuous function such that g(x) >=1 for all x in the interval [-1,1].?

Question

Define f(x)=0 if x=1/n for some n belonging to the natural numbers and f(x)=g(x) otherwise.
The limit as x-->0 of f(x) does not exist, but the limit as x-->a f(x) exists for all a not equal to 0.
Use the Non-Existence Criterion to show that the limits can cease to exist when you alter a function at infinitely many points that are not separated from each other by a positive amount. The separation between pairs of points 1/n and 1/m gets smaller as you make m and n larger. Specifically, prove that for every epsilon>0, there exists N>0 so that if n,m>N then |1/n - 1/m |

Answer 1

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