"We want the domain of the step function:" \qquad f(x) \ = \ [ 2 x - 1 ].
"1) The linear function," \ 2 x - 1, "is defined for all real numbers,"
\qquad \quad x.
"2) The greatest integer function," \ [ x ], \ "is defined for all real"
\qquad \quad "numbers," \ \ x.
"3) So now, in language that should be clear:"
\qquad \qquad \qquad \qquad f( "real" ) \ = \ [ \ 2("real") -1 \ ] \ = \ [ \ "real" \ ] \ = \ "real".
"4) Thus:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad f( "real" ) \ = \ "real".
\qquad \quad "So:"
\qquad \qquad \qquad \qquad \qquad \ f(x) \quad "is defined for all real numbers" \ \ x.
\qquad \quad "Thus, in interval notation, and in set notation:"
\qquad \qquad \quad \qquad "domain of" \quad f(x) \ = \ ( - infty, infty ) \ = \ RR.
"This is our answer."