I am studying microeconomics at the introductory undergraduate level and two related but distinct math-related questions are puzzling me.
First, my textbooks express utility functions as continuous functions by default, but this is puzzling. If the consumer can only consumer in whole number quantities (indeed from what I've seen so far, the solutions for the optimal basket all involve whole number quantities), then surely the utility function should be a discrete function, not continuous? A continuous utility function implies that an optimal basket could be say, $x=8.232324232342, y=3.23942$, for instance.
Secondly, even we if we accept as fact that a utility function can be continuous, how exactly do we interpret the concept of marginal utility at a point? So for instance, given the utility function (single good) $U(X)=X^2+6X+7$, the marginal utility of consuming the 6th good should be $U(6)-U(5)$, because marginal utility is the increase in utility as a result of consuming an additional unit of the good. Instead, marginal utility is, in fact just, $U'(6)$ which is usually not equivalent to $U(6)-U(5)$ (it's not in this case, too). So what's going on here? Am I missing something? I know the former way is called the "arc" measure, and the latter the "point" measure - I understand the arc measure, but I cannot understand the point measure. Doesn't marginal utility always measure the increase in utility arising from consuming one additional unit of the good? The point measure seems to imply $U'(6) = U(6)-U(5.999999999...)$.
Utility is a way to order your preferences between different baskets of goods, and it is continuous since we assume all goods are infinitely divisible. The marginal utility at a point is the increased utility from an extra unit of consumption at the current level of consumption. In the utility framework, you can consume fractions of a unit. It's not all that helpful to think of utility as an absolute magnitude because utility doesn't exist in reality. What's important is the sign a relative magnitude ($U(x) > U(x')$ so I prefer x to x')