I am trying to summarize some definitions regarding the different types of continuity I know, in my own words, and I would like to know if you think they are correct (that is, if they are equivalent to the definitions you know of these terms)
Let $X,Y$ be metric spaces, $(f_i)_{i \in I} \subseteq Y^X$ a family of functions. Let $x \in X$ and $\epsilon > 0$.
Let $D(f_i,x,\epsilon)$ be the suprem of the $\delta \geq 0$ such that
$$ f_i(B(x, \delta)) \subseteq B(f_i(x), \epsilon) $$
Note that if $f_i$ is not continuous in $x$ then $D(f_i,x,\epsilon) = 0$ for some $\epsilon > 0$ (in this case there exists some $\epsilon > 0$ such that there is no $\delta > 0$ such that $f_i$ is continuous at $x$.)
Note also that $\delta = 0$ satisfies the condition for all $f_i, x, \epsilon$
1) If $ D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)$ the function $f_i$ is continuous at $x$
2) If $ D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)(\forall x \in X)$ the function $f_i$ is continuous
3) If $ D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)(\forall i \in I)$ the family $(f_i)$ is continuous at $x$
4) If $ D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)(\forall i \in I)(\forall x \in X)$ the family $(f_i)$ is continuous
5) If $\sup_{x \in X} D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)$ the function $f_i$ is uniformly continuous
6) If $\sup_{x \in X} D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)(\forall i \in I)$ the family $(f_i)$ is uniformly continuous
7) If $\sup_{i \in I} D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)$ the family $(f_i)$ is equicontinuous at $x$
8) If $\sup_{i \in I} D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)(\forall x \in X)$ the family $(f_i)$ is equicontinuous
9) If $\sup_{x \in X, i \in I} D(f_i,x,\epsilon) > 0 \ (\forall \epsilon > 0)$ the family $(f_i)$ is uniformly equicontinuous
I am not even sure if 1) is correct. What do you think?