Since
f
maps the interval [0,1] into the interval [0,1] and we are assuming that
f
(0) does not equal 0,
f
must be greater than 0. But then
g
(0) =
f
(0) - 0 > 0. Similarly since
f
(1) does not equal 1,
f
must be less then 1, so
g
(1) =
f
(1) - 1 < 0.