Charles - an order-preserving map from \$$\mathbb{R}\$$ to itself cannot decrease, so it can't map the real line to itself in a manner like folding a strip of paper. The function \$$f(x) = |x|\$$, thought of as a map from the line to itself, folds the line over.

But notice that \$$f(x) = -x\$$ is not order-preserving and does not fold the line.

If you want to learn MathJax just click on the little gear that will appear when you mouse over someone's comment, and click on "View Source". Then you'll see what they did. For example, in my first paragraph I wrote this:

Charles - an order-preserving map from \$$\mathbb{R}\$$ to itself cannot decrease, so it can't map the real line to itself in a manner like folding a strip of paper. The function \$$f(x) = |x|\$$, thought of as a map from the line to itself, folds the line over.