Weak contraction mapping preserves order?

Question

Suppose I have a function $\varphi$ on the unit interval such that $|\varphi (x)-\varphi (y)|<|x-y|$.

Is it true that if $0 \leqslant c \leqslant 1$ then $\varphi (0) \leqslant \varphi (c) \leqslant \varphi (1)$ or $\varphi (1) \leqslant \varphi (c) \leqslant \varphi (0) $ ?

In particular I am curious whether such a mapping maps the endpoints of a segment to the endpoints of the image segment (The image has to be a segment due to compactness and connectedness)

Any generalizations perhaps?

Answer 1

Take $\phi(x) = {1 \over 2} |x-{1 \over 2}|$. Then $\phi$ is Lipschitz of order ${1 \over 2}$.

We have $0 < {1 \over 2} < 1$ but $\phi(0)> \phi ({1 \over 2}) $ and $\phi(1)> \phi ({1 \over 2}) $.

Answer 2

For more insight, consider $\phi(x) = \frac{1}{2 \pi n} \sin(2 \pi n x)$ for large $n$.