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**zzzhhh** The common definition of uniform convergence is based on sequence, that is, the functions are indexed by natural numbers. But what if the functions are indexed by real number, e.g. $\displaystyle f_t\to f$ uniformly as $\displaystyle t\to t_0$? I guess the definition might be: for any $\displaystyle \epsilon>0$, there is a $\displaystyle \delta>0$ such that $\displaystyle |f_t(x)-f(x)|<\epsilon$ for all $\displaystyle x$ whenever $\displaystyle 0<|t-t_0|<\delta$. My question is: Is it right to exclude the point $\displaystyle t_0$, like we do for ordinary definition of limits?

In addition, Does this type of uniform convergence preserves properties like differential, integral and limits of functions, as does the common case? Thanks!

First, in the ordinary definition of limits, we *don't* exclude $\displaystyle t_0$. The reason why some people do so is usually because they are embarrassed with limits like $\displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1$. The correct way to justify this limit is to write

$\displaystyle \lim_{x\to 0,\ x\neq 0} \frac{\sin x}{x} =1$

and read "the limit of $\displaystyle \frac{\sin x}{x}$ when $\displaystyle x$ tends to $\displaystyle 0$ in $\displaystyle \mathbb{R}\setminus\{0\}$ is 1". It is just because of lazyness, or because the function is obviously not defined at the limit point, that we usually skip the "$\displaystyle x\neq 0$" part.

As for the uniform convergence of functions, you can use the "sequential characterization of limits" to prove everything you suggest: $\displaystyle f_t$ converges uniformly to $\displaystyle f$ as $\displaystyle t\to t_0$ if, and only if, for any sequence $\displaystyle (t_n)_{n\geq 0}$ converging to $\displaystyle t_0$, the sequence $\displaystyle (f_{t_n})_{n\geq 0}$ converges uniformly to $\displaystyle f$. Thus you reduce to a sequence and can apply the usual properties. For instance (under usual hypotheses, like $\displaystyle f_t$ continuous), for any sequence $\displaystyle (t_n)_{n\geq 0}$ converging to $\displaystyle t_0$, $\displaystyle f_{t_n}$ converges uniformly to $\displaystyle f$, hence we have $\displaystyle \int_0^1 f_{t_n}(x) dx \to_n \int_0^1 f(x)dx$. And using the sequential characterization (in a different way), you deduce $\displaystyle \int_0^1 f_t (x) dx\to_{t\to t_0} \int_0^1 f(x)dx$.