The function $\varepsilon_X$ assigns to each point of a given continuum $X$ the closure of the family of all continua that contain $x$ in their interior. We define the class $S(\varepsilon)$ of continua for which the function $\varepsilon_X$ is continuous. On the other hand, we consider the condition $\varepsilon_Y(f(x)) = C^2(f)(\varepsilon_X(x))$ for a mapping $f\colon X \to Y$. This condition defines a class of mappings $M(\varepsilon)$.
During my talk I will investigate classes $S(\varepsilon)$ and $M(\varepsilon)$, and relations between them.