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.
- 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 $V$.
- 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!