Combining IF and SIGN functions in Excel
Try this formula for A starting in the third iteration (row 4):
=IF(SIGN($D3)=SIGN($D2),IF(A3=A2,A3,$B3),IF(A3=A2,$B3,A3))
You can apply the same to C as well.
=IF(SIGN($D3)=SIGN($D2),IF(C3=C2,C3,$B3),IF(C3=C2,$B3,C3))
This uses two levels of conditions; first it checks for sign change in f(m), and then checks whether the midpoint was previously being mapped to the lower or upper bound.
Sample output:
