There may be that you just call it the family of subsets of the domain of $f$ on which $f$ is continuous, named $\mathcal{F}$. But if there is a way to nicely say this, or even better a standard notation, I would like to know it.
Edit to clarify the context of the problem:
I have the following 2 functions:
$$f:\mathbb{R}^n\rightarrow\mathbb{R}$$
$$g:\mathbb{R}\rightarrow\mathbb{R}$$
$g$ is defined as being not continuous in all points of $D\subset\mathbb{R}$
$g(f(x_1,x_2,..x_n))=y_g$
$g(x_1)=y_1$
$g(x_2)=y_2$
...
$g(x_n)=y_n$
Each of $y_1, y_2,.. y_n$ will be part of some intervals of the codomain (let's call them $\mathcal{G_y}$) of $g$ that are the images of $g$ on each of its own intervals on which it is continuous that contain $x_1, x_2,..x_n$ (let's call them $\mathcal{G_x}$. My goal is to formulate a condition to define the subset of functions $f$, for which $y_g$ is guaranteed to be in any of the intervals of $\mathcal{G_y}$.
E.g. for $n=3$ $$f_{mean}(\boldsymbol{x})=\frac{x_1+x_2+x_3}{3}$$ would fail this condition, since I can find a function $$g(x)=\begin{cases}1, && x<5 \\2, &&5\leq x\leq 10 \\3, && x>10 \end{cases}$$ and some input values for
$f(4,4,13)=7$,
$g(7)=2$,
but $g(\{4,4,13\})=\{1,1,3\}$, which does not contain 2.
With each member of $g$ in this example being a constant function, it is easy to say that $2 \notin \{1,1,3\}$, but for a function $g$ with non-constant branches, I would care for $g(f(\boldsymbol{x}))$ to be in one of the branches of $g(\{x_1,x_2,..x_n\})$
An example of a function that would meet my condition would be: $$f_{select}(\boldsymbol{x})=x_1$$
Now if I could also prove that only functions of the type $$f_{select}(\boldsymbol{x})=x_i$$ meet my condition then that would be a nice bonus.
Now from the beginning in plain English.
I want to make it wrong for someone to answer that they're fine if they have one foot in ice and the other in boiling water. Telling me you're cold, is ok. Telling me you're hot is also a correct answer. Telling me you're just fine because you average (or apply some other function on) the temperatures of the 2 feet is a wrong answer.
The equivalent plain English example for non-constant branches of $g$ would be that kinda cold, coldish, super cold, freezing cold, sort of fine, fine, super fine, kinda hot, hottish, super hot, brazing hot, etc. are also valid responses and each set of responses that contain cold, fine, or hot correspond to intervals on which g is continuous.