How to call a function on sets that preserves inclusion?

Question

I am writing a paper, and need to define a function on a real interval that generates sets: $$ f : [0,1] \mapsto E\\ \qquad x \quad \mapsto A $$

($A$ is typically a subset of $\mathbb{R}^n$)

I would like to know if the following property could be named:

Given two sets $A, B$ such that $f(0) = A$, $f(1) = B$ and $A \subseteq B$, then $ \forall x \in [0,1], f(x) \subseteq B$

This is akin to monotonicity but is looser since $f$ can freely increase or decrease over $[0,1]$. It can also be seen as a boundedness constraint: $f$ is bounded by its "right" value.

Is there a name for this property?

Answer 1

If you put $f([0,1])\subset f(1)$, then $f(0)$ must be in $f(1)$. So, you can call this by «$f(1)$ is a superior bound for f»