Problem:The set of continuous functions $\displaystyle f:[0,1] \rightarrow [0,1]$ such that f(0)=0, f(1)=1, and f is increasing. Binary operation is composition of functions.

First off, I'm a little confused about the $\displaystyle f:[0,1] \rightarrow [0,1]$ part. Is that just saying the domain and range of the function is from 0 to 1?

Now, I trying to test this for associativity.

$\displaystyle a*(b*c)=(a*b)*c$

But I started to run into problems. There are only two elements in this set, but I need three to test for associativity, don't I? Or do I just use one twice? But what if one works twice, but the other doesn't?

Regardless, I tried using composition to test for associativity, and got stuck.

$\displaystyle a*b = a(b(x))$ ??

Any help is appreciated.