Ok h(x) = f(x) - x

h(0) = f(0) >= 0 since the range is [0,1] if h(0) = 0 then f(0) = 0 done

suppose then h(0) > 0

h(1)= f(1) - 1 < = 0 again since the range is [0,1]

If h(1) = 0 f(1) = 1 done

if not h(1) > 0

since h(1) >0 and h (0) < 0 then there is a c st h(c) = 0

h(c) = f(c) - c = 0 i.e f(c) = c