Borel-measurable function

Prove that the following function is Borel-measurable function.

f_n(t) = {   [t*2^n]*2^-n                , 0 < t < n,
              
                   n                                 , t > or = to n
                
                   | f_n(t) - t | < 2^-n       , t < n  }


I want a detailed proof. I want to know what one needs to check when proving some function is Borel-measurable function.

There are no typos, the way the function is defined for 0 < t < n is approximation for the function