Partition of $[a,b]\subset\mathbb{R}$

Question

Is it possible to create a numerable partition of $[a,b]$? Because I think that it isn't possible, because the last point of the partition must be $b$. But I use the function defined in $[0,1]$ that: $$ f(t)=\frac{1}{2^{n-1}} \mbox{ if } t\in \left[\frac{1}{2^n},\frac{1}{2^{n-1}}\right] $$ I can't create a partition in order to have a point of the partition in every set where $f(t)$ changes its value. What do you think? Thanks for the help!