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Mathew Lane DrMS V4.0 VST RTAS X32 X64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS V4.0 VST RTAS X32 X64 – DYNAMiCS [deepstatus] Serial Key
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Mathew Lane DrMS V4.0 VST RTAS X32 X64 – DYNAMiCS [deepstatus] Serial Key.
Mathew Lane DrMS v4.0 VST RTAS x32 x64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS V4.0 VST RTAS X32 X64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS v4.0 VST RTAS x32 x64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS V4.0 VST RTAS X32 X64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS v4.0 VST RTAS x32 x64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS V4.0 VST RTAS X32 X64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS v4.0 VST RTAS x32 x64 – DYNAMiCS [deepstatus] Serial Key
Mathew Lane DrMS V4.0 VST RTAS X32 X64 – DYNAMiCS [deepstatus] Serial Key
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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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