Let $(X,d)$ be a metric space, and let $F:X\times[0,B)\to\mathbb{R}$ be a function. My question is the following:
What does it mean for $F(\cdot,t)$ to converge uniformly as $t\to B$?
Equivalently, one may think of $F$ as a family of functions $f_t:X\to\mathbb{R}$ defined by \begin{align} f_t(x):=F(x,t),\qquad\forall x\in X \end{align} continuously parametrized by $t\in[0,B)$. If I am correct, the question is then equivalent to ask about
What does it mean for the family $\{f_t\}_{t\in[0,B)}$ to converge uniformly as $t\to B$?
I ask this question because uniform convergence is usually defined for a sequence of functions $\{f_n\}_{n\in\mathbb{N}}$, and as far as I could find, there is no any reference that has an explicit answer to my questions above. I wonder if this is because such generalization (from a sequence of functions to a continuous family of functions) is in fact easy and just left to the readers.
Bearing this in mind, I thus make the following guess: Choose a sequence $\{t_n\}_{n\in\mathbb{N}}$ in $[0,B)$ such that $t_n\to B$ as $n\to\infty$. Then consider the sequence $f_n:=f_{t_n}$. We say that $\{f_t\}_{t\in[0,B)}$ converges uniformly as $t\to B$ if $\{f_n\}_{n\in\mathbb{N}}$ converges uniformly as $n\to\mathbb{B}$.
I am not that confident to this guess though. One major problem that has to be resolved is to show that the definition is independent of the choice of the sequence $\{t_n\}$. However, even if this is not a big problem, I feel that there is still something lacking.
Any comment, suggestion and answer is welcomed and greatly appreciated. In particular, if there is actually some reference which gives explicitly the definition, please let me know as well.
It means there is a function $G(x)$ such that for every $\epsilon >0$ there exists $\delta >0$ with the property $|F(x,t)-G(x)| <\epsilon$ for all $x$ whenever $B-\delta <t <B$. (This $\delta$ has to be independent of $x$).