0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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}) $.

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.