## Windows Live Writer

In this blog post, I take Windows Live Writer for a test drive.

As a mere blogger wannabe, I can’t say for sure what it is I’d like to see in a blog-posting app. More than likely, the limitations are going to be found in the host software rather than the editing tool. For example, I’d sure like to embed little javascript apps in my posts, but WordPress doesn’t allow it.

Another must is LaTeX, or some other convenient way to put math notation in a post. One can always render with some other tool, and then embed the image, but that’s just too much work.

The third and final must-have is a way to drop code in a post and have it look pretty. I can see right now that there’s a Windows Live Writer  plug-in for this. I suppose that means that WLW will render HTML for syntax coloring, so there’s no need for any special ability on the host.

Categories: Uncategorized

## Test Post

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

Categories: Uncategorized

## 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:

Categories: Uncategorized

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

OK, I think I have a proof of the following:
If $f(a+b) = f(a) + f(b) + 2ab$, then $f(x) = x^2 + mx$ for some real m.

Proof:
Setting $a=0$ and $b=0$ , we have $f(0+0) = f(0) + f(0) + 0$;
in other words $f(0) = 0$.
Next, set $a=x$ and $b=-x$ , which yields $f(x-x) = f(x) + f(-x) - 2x^2$.
Since $f(x-x) = f(0) = 0$, we have $f(x) + f(-x) = 2x^2$, or
$f(x) - x^2 = -[f(-x) - (-x)^2]$.
Let $g(x) = f(x) - x^2$ (the left side)
Then we see that $f(x) = x^2 + g(x)$ and $g$ is an odd function, i.e. $g(x) = -g(-x)$.
Now,
$f(a+b) = (a+b)^2 + g(a+b) = a^2 + b^2 + 2ab + g(a+b)$
but also
$f(a+b) = f(a) + f(b) + 2ab$
Setting equal the right sides of the above two equations and simplifying,
we get
$g(a+b) = g(a) + g(b)$,
which is the definition of linear function with $g(0) = 0$. (y-intercept = 0)

Categories: math