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I'm stuck with this proof I just can't get my head around and I would really appreciate any sort of help. The problem is as follows:

Problem: Let $f$ be a function defined in the neighborhood of the point $x=x_{0}$. It is given that for all $\epsilon>0$ there exists a number $\delta>0$ such that for all $x, y$, if $0<|x-x_{0}|<\delta$ and $0<|y-x_{0}|<\delta$, then $|f(x)-f(y)|<\epsilon$. Show that the limit of $f$ exists at $x_{0}$.

Thanx.

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