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$u(x_1,x_2)=|x_1 −2|+x_2$

Is this function continuous?

PS: The continuity theorem I use is this: Whenever for any $x^n$, $n \in N$ with $x^n \to x$ (i.e. $\lim_{n \to \infty} x^n = x$) and for any $y^n$, $n \in N$ with $y^n \to y$ with the property that $x^n$ is weakly preferred to $y^n$ for all $n \in N$, it must be that $x$ is weakly preferred to $y$.

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    $\begingroup$ I do not understand your definition of continuous, and using LaTeX might help readability, but mathematically $u(x_1,x_2) = |x_1-2| +x_2$ is certainly continuous, though not differentiable when $x_1=2$ $\endgroup$ – Henry Oct 1 '19 at 22:00
  • $\begingroup$ I think maybe you mean this definition? In this case you might want to edit the question to reflect this. en.wikipedia.org/wiki/Debreu_theorems#Statement $\endgroup$ – Art Oct 2 '19 at 16:39
  • $\begingroup$ Perhaps you meant to ask whether the given function represents a continuous preference? In that case, the answer is pretty obvious given Henry's comment above. $\endgroup$ – Herr K. Oct 2 '19 at 19:56

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