# Is this utility function continuous?

$$u(x_1,x_2)=|x_1 −2|+x_2$$

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 $$y$$.

• 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 $x_1=2$ – Henry Oct 1 '19 at 22:00
• 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 – Art Oct 2 '19 at 16:39
• 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. – Herr K. Oct 2 '19 at 19:56