Why can utility functions be continuous, and what does this imply for marginal utility?
Utility functions can be continuous because quantitites can be continuous. (Think liters instead of bottles of wine or kilos instead of loafs of bread.) But even if quantities are discrete; whenever a unit is reasonably small (grains of salt: yes; cars: no) it is just way more convenient to work with smooth functions than with discrete ones, since the former allow you to calculate derivatives.

Introductory textbooks often use discrete quantities ("additional utility of consuming the next unit") to define marginal utility, since this is more intuitive. However, as soon as you have smooth utility functions, you better use the derivative of utility instead. Think of the point measure as the limit of the arc measure as the increase in quantity goes to zero. (That's more or less how the derivative is defined.) If quantities are discrete but very small and your utility function is reasonable, then the two measures are almost identical anyway. As an example, if your utility function is defined over $x$ = the number of grains of salt on your chips, then, using e.g. $u(x)=1-(x/100-1)^2$, your $MU$ at $x=100$ is $0$ if you use the derivative and $-0.0001$ if you just calculate $u(101)-u(100)$. To find the optimal quantity, setting $MU=0$ is of course easier than listing 200 discrete values and searching for the maximum.