how to convert Piecewise function to normal function of one line like

f(x)={ 2:x=0 ;; 1 : x>0 }

f(x)=floor( (x+2) / (x+1) )

g(x)={ a : x=0 ;; b*a : x>0 }

g(x)=a + (a-1)*b*( 2-f(x) )

As Rect( (x-a)/b ) = H(x - a - b/2) - H(x - a + b/2) then your expression is exactly the same as that given above by @Christoforos. Hence, it still does not satisfy the BC at at x = 0; x = 0.5 and x = 1.
If you define G(x) = H(x) + H(- x²) then
f(x) = G(x) + G(x-1) - 2G(x-0.5

As Rect( (x-a)/b ) = H(x - a - b/2) - H(x - a + b/2) then your expression is exactly the same as that given above by @Christoforos. Hence, it still does not satisfy the BC at at x = 0; x = 0.5 and x = 1.
If you define G(x) = H(x) + H(- x²) then
f(x) = G(x) + G(x-1) - 2G(x-0.5