Definition 11 taken from Hardouin et al. 2018.
Definition 11 (Continuity). A mapping $\Pi$ from a complete idempotent semiring $D$ to a complete idempotent semiring $C$ is lower semi-continuous (denoted l.s.c.) if for all finite or infinite sets $X \subseteq D$,
$$\Pi\left(\bigoplus_{x\in X} x\right) = \bigoplus_{x\in X} \Pi(x),$$
and it is upper semi-continuous (denoted u.s.c) if for all finite or infinite sets $X \subseteq D$
$$\Pi\left(\bigwedge_{x\in X} x\right) = \bigwedge_{x\in X} \Pi(x).$$
It is continuous if it is both l.s.c. and u.s.c.
I struggle to understand how this relates to the definition given in analysis or this definition , we can take the $(\overline{\mathbb{R}},\oplus,\otimes)$ dioid for $C$ where $\otimes$ is addition and $\oplus$ is maximization. This definition doesn't need limits to exist, and doesn't define them either. I just don't get how it intuitively relates to this. I just don't see how this makes a function continuous over dioids.
