The family of continuous, bounded (in $[0,1]$) functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows:
\begin{equation} f(y) = \begin{cases} 0, &\hat{l}(y)<\rho \\ \kappa(y), & \hat{l}(y)=\rho\\ 1, & \hat{l}(y)> \rho \end{cases} \end{equation}
$\rho\in(0,1)$ is a real number, $\kappa$ is a function from $I$ to $[0,1]$ such that $f$ is still continuous, $\hat{l}$ is a positive function from a set of functions $\mathcal{L}$ on $I$, for which it is known that $\mu(\{y:\hat{l}(y)<\rho\})>0$, $\mu(\{y:\hat{l}(y)>\rho\})>0$ and $\mu(\{y:\hat{l}(y)=\rho\})>0$ for Lebesgue measure $\mu$.
Question: Is the family of functions $\mathcal{F}$ equicontinuous?
It seems that $$||f||_\infty =\max_{y\in I} f(y)$$ is true and is equal to $1$. Since $f$ is bounded I would guess that $\mathcal{F}$ would be compact w.r.t. the infinity norm.
If $\mathcal{F}$ is compact, then it must be also equicontinuous but I am not sure if It is really compact.
If the answer would be negative is it possible to take $\kappa$ as an increasing function of $l$ to get a positive answer for
\begin{equation} \hat{l}(y) = \begin{cases} l(y)/a, &l(y)<a \\ \rho, & a\leq l(y)\leq b\\ l(y)/b, & l(y)>b \end{cases} \end{equation}
$a<1$, $b>1$ are some numbers and $l=g_1/g_0$ is the ratio of two non identical density functions on $I$.
I may misunderstand the definition of $\mathcal{F}$. But to me, it sounds like I can choose $\rho=1/4$ and
$$l(x) = \begin{cases} x & x \in [0,1/4] \\ 1/4 & x \in [1/4,3/4] \\ 1/4 + (x-3/4) & x \in [3/4,1] \end{cases}$$
Now on $[1/4,3/4]$ I can arbitrarily pick $\kappa$ as long as $\kappa(1/4)=0$ and $\kappa(3/4)=1$. But that means I can choose $\kappa$ as restricted to $[1/3,2/3]$ completely arbitrarily, and $C^0([1/3,2/3])$ is not equicontinuous.
But I think I misunderstand the definition, because under this interpretation $\| f \|_\infty$ can be arbitrarily large. (For example, with the $l$ above, the intermediate $\kappa$ can be $M(4x-1)+(1-M)(4x-1)(4x-2)/2$ for any $M>0$.) I think there is a hypothesis either implicit or not stated (perhaps that $f$ are increasing or something).