Let $X$ be a Polish space and $f : X \to \mathbb{R}$ an arbitrary function. In The limit inferior of Borel functions I showed that a certain function $f$ has the following property:
For each $a \in \mathbb{R}$, the set $f^{-1}((-\infty, a]) = \{x \in X : f(x) \le a\}$ is analytic in $X$.
Is there a name for this property of $f$?
Of course, if the sets $\{x \in X : f(x) \le a\}$ were Borel instead of analytic, then $f$ would simply be a Borel function. But clearly a function with this property need not be Borel (consider $f = 1- 1_{A}$ where $A$ is analytic but not Borel.
Replacing $\le$ with $<$ in the definition (giving an open half-line instead of closed) would not change anything, since countable unions and intersections of analytic sets are analytic. But replacing $\le$ with $\ge$ would make a difference, since analytic sets are not necessarily coanalytic. So I might expect the name to have "upper" or "lower" in it somewhere.
Any elementary properties of this class of functions would also be interesting. For example, it is easy to see it is closed under countable supremum and infimum, and not closed under negation.
Finally, it could also be interesting to consider replacing $\mathbb{R}$ with some other ordered Polish space (e.g. $\mathbb{Z}$, $2^\omega$, $\omega^\omega$, etc.).