Are there functions that can be defined on $\mathbb{R}^2$, are metrics on the whole x-axis (i.e, the set of all points whose second coordinates are $0$), but not metrics on $\mathbb{R}^2$ as a whole? I am especially wondering about the existence of the ones that break the triangle inequality.

More generally, are there functions that can be defined on $\mathbb{R}^2$, but are metrics only on a few straight lines or some other subsets of $\mathbb{R}^2$, but not on $\mathbb{R}^2$ as a whole?



I guess some piecewise functions may fit the description. For example, we let $ d = |x - y| $ only on $[-1, 1] \times [-1, 1]$, but let $ d = (x - y) ^ 2 $ elsewhere on $\mathbb{R}^2$. However, I was looking for "non-piecewise" functions.


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You can take $d\bigl((x,y),(z,t)\bigr)=|x-z|+(y-t)^2$. The restriction to $\{(x,0)\mid x\in\Bbb R\}$ is the usual distance there. But\begin{align}d\bigl((0,0),(0,1)\bigr)+d\bigl((0,1),(0,2)\bigr)&=2\\&<4\\&=d\bigl((0,0),(0,2)\bigr).\end{align}


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