Calculate the normal cone of a set of increasing functions

Let $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ continuous and increasing}\} \subset C[0,1]$.We can show that the set X is closed and thus compact, and also convex. (Roughly speaking, X can be regarded as a set of continuous cdfs)

We define the normal cone to $X$ at $f \in X$ as $N(f;X):=\{\mu\in (C[0,1])^{*} \mid \langle \mu,g-f\rangle \leq 0, \forall g\in X\}$.

For this normal cone $N(f;X)$, I have the following observations:

1. $N(f;X)=\{0\} $ for $f\in \mathbb{int}X$.

2. By taking $g=0^{[0,1]}$, we must have $\int_{0}^1 f \mathbb{d}\mu\geq0 $

3. By taking $g=1^{[0,1]}$, we must have $\int_{0}^1 (1-f) \mathbb{d}\mu\leq0 $

My question is: how to calculate the $N(f;X)$ in the closed-form expression? I cannot find any reference for this type of question.

Furthermore, what if we do not require the functions in $X$ to be continuous (e.g., $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ uniformly bounded and increasing}\} \subset L^1([0,1])$)?

Thank you very much!