General definition of uniform convergence

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!