PURE MATHS function 一問?

Consider g(x) = f(x) - x
g(0) = f(0) - 0 = f(0) >=0 (Since 0<= f(0) <=1)
g(1) = f(1) - 1 <=0 (Since 0<= f(1) <=1)
Case 1: f(0) = 0 or f(1) = 1
In this case, g(0) = 0 or g(1) = 0
Hence there is at least one root for f(x) = x in [0, 1]
Case 2: f(0) not equal to 0 and f(1) not equal to 1.
In this case, g(0) >0 and g(1)<0
As f is a continuous function on [0, 1], then g(x) = f(x) - x must also be a continuous function.
As g(0) is greater than 0 and g(1) is smaller than 0, there must be at least one point on [0, 1] equal to 0.
i.e. There is at least one root for the equation f(x)=x in the interval[0,1].
Combining two cases, there is at least one root for the equation f(x)=x in the interval[0,1].