Why are indifference curves (often) of infinite length?

Question

Indifference curves are often of infinite length.

Is this implied by monotonicity or non-satiation?

If not, what is/are some condition(s) that are sufficient for indifference curves to have infinite length?

More generally, indifference curves are almost always manifolds without boundary. Which means that the curves don't have endpoints. What property of the utility function guarentees this?

quarague · Answer

If you are using the definition of indifference curve on wikipedia then they can be of finite or infinite length. For example if you are completely indifferent between the two goods, the indiffernce curve would be of the form $x+y=const$ which is of finite length (in the positive quadrant).

A necessary criteria for infinite length would be that having $0$ of one good provides you with zero utility, so that any combination of a little bit of both has a higher utility than an arbitrary quantity of one good and zero of the other good.

Edit in response to comment: Indifference curves come in families, one can consider the family of indifference curves $xcdot y = const.$, they satisfy the definition, are of infinite length and if either $x$ or $y$ is equal to zero, then the utility is equal to $0$ as well.

Why are indifference curves (often) of infinite length?

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