I am given the true of false question: If f(x) is continuous and 0<=f(x)<=1 for all x in the interval [0,1], then for some number x f(x)=x. This seems intuitively true, but I'm not sure. All help is greatly appreciated.

Answer 1

Consider g(x) = f(x)-x
g(0) = f(0)-0 >= 0
g(1) = f(1)-1 <= 0

since g is continuous, either g(1)=0 or g(1) < 0.

If g(1) < 0, the Intermediate Value Theorem tells us that at some point c in the interval, g(x) = 0