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Please, fill out the form to report inappropriate content. Note: You will be temporarily banned from posting as a moderator. If you have already registered with us, please login to the site using the button in the top-right of this page to update your download link.Q:

Show that $f$ is not a bijection

Let $(X,d)$ a metric space and $f: X \to X $ a $\mathcal{C}^1$ function such that $f
eq Id_X $ , define $d_1 : X \times X \to \mathbb{R}$ by $d_1(x,y)=|f(x)-f(y)|$. Show that $d_1$ is not a metric.

I have already proved that it is symmetric and satisfies the triangular inequality. I also have done this problem in $\mathbb{R}^2$ using Baire theorem but I don’t know how to do it in $\mathbb{R}^1$

A:

Hint:

For $x,y\in X$ and $a\in [0,1]$, $$d_1(x,y) = \frac{1}{2}\left|f(x) – f(y)\right|.$$
If $f$ is a bijection, $d_1$ is the same thing as the $d$ metric.

Note that $|f(x)-f(y)|\in[0,1]$, so $d_1$ is only a non-reflexive metric.

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