In this blog post, I take Windows Live Writer for a test drive.
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.
This is a test of Windows Live Writer. This is only a test.
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 .
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:
OK, I think I have a proof of the following:
If , then for some real m.
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. .
Setting equal the right sides of the above two equations and simplifying,
which is the definition of linear function with . (y-intercept = 0)