For a compact interval $[0,1]$, we divide it into $N^{1/3}$ subintervals with length $N^{-1/3}$. Define a map $\phi: [0,1]\mapsto [0,1]$ maps each subintervals to its center.
For example, let $X\sim Uniform[0,1]$ and $N=1$. Then there is only interval $[0,1]$ and we maps each point to $1/2$. If $X=0.1$, then $\phi(X)=0.5$.
For general $N=1000$, each subinterval with length $1000^{-1/3}=0.1$. How to find a map such that $ \phi(X) $ is equal to the center of that subinterval?
Like if $X=0.34$, we need $\phi(0.34)=0.35$ which is the center of $[0.3, 0.4]$.
I found an approximate way that takes $\phi(X)=N^{1/3}\times [N^{1/3}X]$. But this is not very properly, especially for $N=1,2,3,4,...,10$. For example, if $N=1000$, $X=0.57$, then $$ \phi(0.57)=1000^{1/3}\times [1000^{1/3}0.57]=0.6 $$ but not the median of $[0.5,0.6]$ which should be $0.55$.
Question: Can we find a proper $\phi$?