I am thinking the following ``continuity'' concept:
Let $f:X\rightarrow Y$ be a function where $X$ and $Y$ are two topological vector spaces. For every $x\in X$, $f(x)\in\overline{co}\{limit\ points\ of \{f(x_n)\}:x_n\rightarrow x\}$.
$\overline{co}$ means closed convex hull. Thus, while $f$ may not be continuous at $x$, $f(x)$ is the mixture of surrounding limit points.
For example, suppose $f:[0,1]\rightarrow [0,1]$. Assume $f$ is continuous everywhere except for $x=a$. Then I require that $f(a)$ lies in between $\lim_{x\uparrow a}f(x)$ and $\lim_{x\downarrow a}f(x)$.
Is there a formal definition or terminology for this type of ``continuity''? Does anyone know any reference? or can provide me some key words to search? Thank you very much!
There is a notion of convexity spaces, which are sets $X$ with a distinguished collection of sets (like a topology or a measurable space is) $\mathcal{C}$ which are called convex, and that obey the 2 convexity axioms:
if $\mathcal{C'} \subseteq \mathcal{C}$, then $\cap \mathcal{C'} \in \mathcal{C}$, so the "convex sets" are closed under arbitrary intersections.
$\mathcal{C}$ is closed under directed unions: if $\mathcal{C'} \subset \mathcal{C}$ is directed (for any $C_1, C_2 \in \mathcal{C'} \exists C_3 \in \mathcal{C'} : C_1 \cup C_2 \subseteq C_3$) then $\cup \mathcal{C'} \in \mathcal{C}$.
Examples include vector spaces with the usual notion of convex sets, or metric spaces with convexity defined from the "inbetween" relation, or sets with a so-called mixer function etc. One can define the notion of a convex closure, (so convex hulls exists) and the flavour of the topic is quite "topological".
The notion you propose seems similar to a sort of analogue of convexity preserving function, but you mix it with topology, using sequences. Maybe look into this field for more ideas. It might be known already.
See the "bible" of this field "Theory of Convex Structures" by my old teacher M.L.J. van der Vel (originally a topologist BTW).