Hint (for 1): Try the mean value theorem.
I am not sure how to address the following two continuity problems, they are quite similar to one another.
1) Suppose that g(x) is a continuous function on [0,1] and that 0<=g(x)<=1 for all x ∈ [0,1]. Show that there is a value of x in [0, 1] where g(x) = x.
2) Suppose that g(x) is a continuous function on [0,2] with g(0) = g(2). Show that there is a value of x in [0,1] such that g(x) = g(x+1).
Thank you in advance for the help.
You probably know the intermediate value theorem: if f(x) is a continuous function on an interval [a,b], then for every c between f(a) and f(b), there is an x in [a,b] such that f(x)=c. For (1), if g(0)=0 or g(1)=1 you have your value. If g(0)<0 and g(1)>1, then consider the function g(x)-x. For (2), if g(1)=g(0) you have your value. So g(1) is either greater than or less than g(0), so you have two cases, and in both cases you look at g(x+1)-g(x) for x in [0,1].