Subgradient everywhere terminology

Suppose I have a function $h$ and a function $f$. The function $f$ is non-differentiable and thus, does not have a gradient. However, the function $h$ belongs to the subgradient of $f$ at every $x$, i.e., $h(x) \in \partial f(x)$ for all $x$.

Does this function $h$ has a name? Would it be appropriate to call it simply a "subgradient" of $f$ without specifying at which $x$ it a subgradient?

Such an $h$ is called a selection of the subdifferential.

Given a set-valued function $A \colon X \rightrightarrows Y$, a function $a \colon X \to Y$ is a selection of $A$, if $$ a(x) \in A(x) \qquad\forall x \in X.$$

Typical usages are "measurable selections" and "continuous selections".