are greatest integer function and step up function same thing or different thing.If same then please explain all proofs of greatest integer function's properties and if different then whats the difference between them?

 The unit step function is just a piecewise function with a jump discontinuity at t = a and its graph can be shown as, 

and it can be defined as, 
$u_c\left(t\right)=\left\{\begin{array}{l}0 if t<c\\ 1 if t \ge c\end{array}\right.$

Now the greatest integer function, 

The function f :  R → R defined by

f (x) = [x], x ∈ R, assuming the value of the greatest integer less then or equal to x is called greatest integer function.

Example : The greatest integer less than or equal to 3.9 is 3

∴ [3.9] = 3