Kerisos, Chapter 6, Mishnah 4. The Mishnah discusses how Yom Kippur fits into the laws of bringing korbanos: does the atonement effected by Yom Kippur exempt a person from bringing his korban, and how does this work?
The text of the Mishnah:
"Chayvei chataos va'ashamos vadai she'avar aleihem Yom Hakippurim - chayavim lehavi le'achar Yom Hakippurim" - one who is obligated to bring a chatas or an asham because he knows with certainty that he committed the transgression is not exempt from his obligation even though Yom Kippur has passed. Yom Kippur indeed atones, but it does not have the power to exempt one from bringing the korban, which is a condition for complete atonement.
"Chayvei ashamos teluyim - peturim" - one who became obligated in an asham talui, since he is in doubt whether he sinned or not, is exempt after Yom Kippur has passed. The asham talui comes to protect against the consequences of a sin that may or may not have been committed, and once Yom Kippur arrives, the person is purified from it.
The source of the law in the verse:
The basis for this is written in Vayikra 16:30: "Ki bayom hazeh yechaper aleichem letaher etchem mikol chatoteichem lifnei Hashem titaharu" - for on this day He shall provide atonement for you to purify you; from all your sins before Hashem shall you be purified. We must analyze: what do the words "before Hashem" come to add?
Chazal learned from here that the transgressions that only the Holy One, Blessed is He, knows about are those that are "before Hashem," since people do not know about them, neither the sinner himself nor anyone else. Yom Kippur atones for these transgressions on its own, without any additional requirement. This is exactly the case of one who becomes obligated in an asham talui: he does not know if he committed the transgression, and no one else knows it, and therefore it is "before Hashem," and the atonement is effected by the power of Yom Kippur alone.
In contrast, for any transgression that has a defined obligation - bringing a chatas or an asham, or lashes that Beis Din is commanded to administer - Yom Kippur alone is not sufficient, and it is still incumbent upon the person to fulfill the rest of his obligations.
This is not a great surprise. If a person broke his friend's window, and after Yom Kippur claims that he is not obligated to compensate him because he achieved atonement on Yom Kippur - they will answer him that this is not the point: the stain on the soul may indeed have been cleansed if he repented, but the obligation to appease his friend and repair the damage remains in place. And so it certainly is regarding the obligation to bring a chatas and an asham.
Which part of Yom Kippur atones?
The second part of the Mishnah deals with this question: is there a specific moment that atones, the beginning of the day or its end? The answer is that Yom Kippur is somewhat similar to a mikvah - even a momentary immersion is effective for purity, and there is no need for a specific amount of time. Each and every part of the day atones.
For illustration: a doctor instructed a person that he must eat on Yom Kippur due to a danger to his life. That person went to the refrigerator and took a food that he thought was kosher, but in fact there were two pieces there, one of chelev (forbidden fat) and the other permissible, and he does not know which of them he ate. This is a doubtful transgression that obligates an asham talui - except that the matter occurred on Yom Kippur itself.
Regarding this, the Mishnah says: "Mi sheba al yado safek aveirah beYom Hakippurim, afilu im chasheichah - patur, shekol hayom mechaper" - one who commits a doubtful transgression on Yom Kippur, even at nightfall, is exempt, for the entire day atones. Even if the doubt arose in the final seconds of Yom Kippur, at sunset, and only a single second of the day remained to atone for him - he is forever exempt from bringing an asham talui, since each and every moment of the day effects complete atonement.
From this we learn that a person who passes away one minute into Yom Kippur - gains Yom Kippur, and similarly, if someone stumbles in a sin in the final second of the day - Yom Kippur does its work. Of course, it can only accomplish what it is capable of accomplishing; but as long as the person repents as required of him, and as long as he believes that Yom Kippur atones - which is also a condition in its own right - Yom Kippur carries out its actions of atonement and purification at every single moment.