Proving D is onto
Let be the collection of all functions whose domain is and differentiable on the domain. Also, let be the
collection of all functions whose domain is and continuous on the domain.
I know that I need to prove for any , for some but I dont know how to go about writing the proof. Can anyone help me at least get a start on it? I'm sure I can figure it out with the proper hint.
Also, every example I find online, has equaling some function (i.e. ). In my case, I dont have a specific function so that is why I am stuck and confused.
You are correct. My apologies. It should be in R and not in R^2. I also believe that the below is what I got using what you gave me. However, I guess I still dont see how exactly it proves it is "onto" and would like a worded explanation if you could. Perhaps it just seems too simple or perhaps my definition of what onto means is still blurry?
A function is "onto" if and only if, for every , there exist such that f(v)= u.
My point is that if u is a continuous function on R, then any of its anti-derivatives is a differentiable function, v, such that Dv= u.