Test Post

This is a test of Windows Live Writer. This is only a test.

A Problem I Just Made Up

Show that has at least one solution for all real ,.

Let be the standard inverse of , continuous on the entire real line with range .

If , is easily seen as a solution, so from here on assume .

Define

    

Since

and ,

by the Intermediate Value Theorem there is a value in between for which .

Let . Because , we also have .

We verify that is a solution to the original equation:

If f(a+b) = f(a) + f(b) + 2ab, what is f?

OK, I think I have a proof of the following:
If , then for some real m.

Proof:
Setting and , we have ;
in other words .
Next, set and , which yields .
Since , we have , or
.
Let (the left side)
Then we see that and is an odd function, i.e. .
Now,

but also

Setting equal the right sides of the above two equations and simplifying,
we get
,
which is the definition of linear function with . (y-intercept = 0)