First, wanting to achieve "financial independence" does not imply a discontinuous utility function.
Suppose it takes $X to reach financial independence. Let's say that when I achieve $X, I have utility Y. It is possible that as I approach $X from the left, my utility continuously approaches Y.
In fact, I argue that's what happens for most people who have a FIRE goal — they get happier and happier until they reach their goal. (It's true that once they reach their goal and experience retired life, they decide that retirement's not all that it's cracked up to be, but I argue that's because their utility has dropped because of another variable has changed: they have stopped working.)
In other words, "wanting to not have to work again" and "having a continuous utility function" do not contradict each other. And the people who are dedicated to saving enough money to not have to work (FIRE) almost always have utility functions that are continuous wrt. money!
> when they talk about this stuff.
Again, people might say that they have a discontinuous utility function, but talk is cheap. Real economists (TM) measure utilities through examining a consumer's revealed preferences because what people say doesn't always reflect what people do.