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There is a result in real analysis:

$f:\mathbb{R} \to \mathbb{R}$ has a limit at $x_0 \in \mathbb{R}$, if and only if for any $\epsilon >0$, there exists $\delta >0$, such that whenever $0<|x-x_0| < \delta$ and $0<|x'-x_0| < \delta$, we have $|f(x)-f(x')| < \epsilon$.

I was wondering if it can be generalized to the case when $f: X \to Y$ is between two uniform spaces $X$ and $Y$? Following is my attempt.

  1. Is there a concept of Cauchy convergence of $f$ at $x_0 \in X$? Following is my guess. I call $f$ Cauchy converges at $x_0 \in X$, if for any entourage $V$ of $Y$, there is an entourage $U$ of $X$ such that for every $x$ and $y$ in $U[x_0]$, $(f(x), f(y))$ is in $V$.
  2. So convergence of $f$ at $x_0$ implies Cauchy convergence of $f$ at $x_0$. But when is its converse true? When both domain $X$ and codomain $Y$ are complete?

If easier for discussion, you may restrict to the case when $X$ and $Y$ are metric spaces, although the questions are for uniform spaces.

Thanks and regards!

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In the stated result taking $x = x_0$ or $x' = x_0$ is disallowed. Not in your proposed equivalent? Note also that $x,y \in U[x_0]$ for some entourage $U$ is just taking $x,y$ in any arbitrary basic neighbourhood, right; so it's not really an entourage condition, so sort of mixed topology-uniformity. And lastly, what is your definition (in purely topological terms) of $f$ having a limit at $x_0$ when $f$ is a function between topological spaces? –  Henno Brandsma Feb 21 '12 at 21:31
    
@HennoBrandsma: Thanks! (1) Will it work if I add to my definition of Cauchy convergence that both $x$ and $y$ are not $x_0$? (2) Do you think that Cauchy convergence is defined for a mapping from a topological space to a uniform space, instead of from a uniform space to a uniform space? (3) The definition of $f$ between topological spaces having a limit at $x_0$ is: for any nbhd of the limit, there is a punched nbhd of $x_0$, such that its image is a subset of the nbhd of the limit. –  Tim Feb 22 '12 at 1:23

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