Let $x$ be a real number, $x \in [0,1]$. Suppose a system can only provide a noisy signal about the value of $x$, given the granularity allowed by the system, $N \in \mathbb{N}^*$.
I'm looking for an equation/formula to round up $x$ given the number of steps $N$ existing within $[0,1]$. See example below.
Example
$x = 0.3$
$N = 2$, meaning that within the range [0,1] the system can only take either the value of $0$ or $1$. Given this granularity, the system should "round up" $x$ to $0$.
$N=5$, the system can now take the values $0$, $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, $1$. Given this new granularity, $x$ should become $\frac{1}{4}$.
EDIT: Preferably not using floor or ceiling functions, i.e.
$$\frac {\lfloor x \cdot N \rfloor}{N -1}$$
You can use the following function for calculating floor:
$$f(x)=x-\frac{1}{2}-\frac{\arcsin(\sin(\pi(x-\frac{1}{2})))}{\pi}$$
Then, for your purpose, you can simply use:
$$\frac{f(xN)}{N-1}$$
If it helps, then you can also use the following function for calculating ceiling:
$$f(x)=x+\frac{1}{2}+\frac{\arctan(\tan(\pi(-x-\frac{1}{2})))}{\pi}$$
If memory serves correctly, these functions yield incorrect results for negative integers.
Your question, however, implies that you are interested only in positive real numbers...