I must be missing something... I am currently reading the book by Burago,Burago and Ivanov, A Course in Metric Geometry, in which it was stated that the map $d_{\log}: \mathbb{R}^2 \rightarrow \mathbb{R}^+$ defined by $d_{\log}(x, y) = \log|x-y|$ is a metric on $\mathbb{R}$.
It seems to me that this map is not only undefined for when $ x = y$, but there also seems to be the issue that it returns negative values for when $|x-y| < 1$ and fails the positivity condition for when $y = x + 1$.
I was perhaps thinking that the function would require some redefinition for various pairs $(x,y)$; with this being the case, however, I still find it strange that the authors would go so far as to make that initial claim without any disclaimer. Could anyone help clarify this situation?
Thanks in advance!
This is obviously not a metric for the reasons you described.
If you're reading the online version, the problem lies there -- the actual book uses $d_{log}(x,y)=\log(|x-y|+1)$, which is indeed a metric: symmetric, non-negative (and zero iff $x=y$) and you can check the triangle inequality on your own.