Gemara

Suggestion: Perhaps Hoche'ach obligates rebuking once, and Tochi'ach teaches a second time (but no more)!

Rejection (Rava): Hoche'ach connotes even 100 times. Tochi'ach teaches that even a Talmid must rebuke his Rebbi.

Erchin 16b (Beraisa): If the transgressor did not accept the rebuke, one must rebuke him again. We learn from "Tochi'ach" - in any case.

Suggestion: This is even if he will blush from embarrassment!

Rejection: "V'Lo Sisa Alav Chet".

Question: To what extent must one rebuke?

Answer #1 (Rav): One must rebuke until the point that the transgressor is ready to hit him.

Answer #2 (Shmuel): One must rebuke until he is ready to curse him.

Answer #3 (R. Yochanan): One must rebuke until he gets angry.

Tana'im argued like the Amora'im. In a Beraisa, R. Eliezer, R. Yehoshua and Ben Azai say like Rav. Shmuel and R. Yochanan, respectively

All expound "va'Yichar Af Sha'ul bi'Yehonason... va'Yatel Sha'ul Es ha'Chanis Lehakoso." The three opinions learn from "Lehakoso", "l'Voshtecha ul'Voshes Ervas Imecha"; and "va'Yichar Af Sha'ul", respectively.

Question (against the third opinion): Even after Sha'ul was angry, Yonason rebuked him to the point of hitting and cursing!

Answer: Yonason loved David exceedingly, therefore he rebuked his father (for wanting to kill David for no reason) more than one is required to.

Shabbos 55a: R. Zeira asked R. Simon to protest against the household of the Reish Galusa. Simon said that it would not help.

R. Zeira: Still, you should protest (you do not know for sure)!

148b - Question (Rava bar Rav Chanan): A Mishnah forbids, on Yom Tov, to clap hands on each other or over the heart, or dance (lest one fix a musical instrument). People do this, and we do not protest!

Counter-question (Abaye): Rava taught that one should not sit even with the Lechi, lest something roll away, and he will retrieve it. People set jugs down and sit there, and we do not protest!

Answer to both questions (Abaye): We are silent because (they will not listen, and) it is better that Yisraelim transgress b'Shogeg than b'Mezid!

This is even for an Isur mid'Oraisa. Mid'Oraisa, one must add to the fast of Yom Kipur (before nightfall). Women eat until night, and we do not protest!

Yevamos 65b (R. Ilai): Just like it is a Mitzvah to say something that will be heeded, it is a Mitzvah not to say something that will not be heeded.

(R. Aba): This is an obligation - "don't rebuke a scoffer, lest he hate you. Rebuke a Chacham, and he will love you."

Rishonim

Rif and Rosh (17a and Yevamos 6:21): Just like it is a Mitzvah to say something that will be heeded, it is a Mitzvah not to say something that will not be heeded. "Don't rebuke a scoffer, lest he hate you."

Rif and Rosh (Beitzah 16b and 4:2): We do not protest against people who clap hands or dance on Yom Tov because it is better that Yisraelim transgress b'Shogeg than b'Mezid. This is only if we are sure that they will not listen. If it is possible that they will listen, we protest. This applies even to a mid'Oraisa transgression. Mid'Oraisa, one must add to Inuy (begin fasting before nightfall), yet some eat until night, and we do not protest!

Rambam (Hilchos De'os 6:7): If one gave rebuke and it was not accepted, he must rebuke a second and third time. He must rebuke constantly until the transgressor hits him and says that he will not listen.

Question (SMaG): Why does the Rambam rule like Rav against R. Yochanan (who says that it suffices to rebuke until the transgressor gets angry)?

Answer #1 (Kesef Mishneh): R. Yochanan holds like Ben Azai, and surely the Halachah does not follow Ben Azai against R. Eliezer and R. Yehoshua (the Rebbi'im of Ben Azai's Rebbi, R. Akiva). The Halachah follows Rav against Shmuel. Alternatively, the Rambam is stringent to rule like Rav.

Note: The Gemara connotes that the Amora'im had not heard the opinions of the Tana'im in the Beraisa. The Kesef Mishneh must say that R. Yochanan would not have ruled like Ben Azai had he known that R. Eliezer and R. Yehoshua disagree with him.

Objection (Lechem Mishneh): Likewise, we should say that the Halachah does not follow R. Eliezer, for he is Shemuti!

Answer #2 (Lechem Mishneh): The Halachah does not follow R. Yochanan because the verse mentions hitting and cursing. The answer given is inadequate.

Hagahos Chavos Ya'ir (on the Rif, 5): Ben Azai holds that Yonason rebuked more than is required due to his love for David. If so, what is the source to require rebuking to the point of anger?

Answer #3 (Gra OC 608:2 DH Ad): The Halachah does not follow R. Yochanan against Rav and Shmuel. (Both of them say that Yonason did not go beyond the letter of the law.)

Sefer ha'Chinuch (239): One must rebuke until the point of hitting, i.e. the transgressor is on the verge of hitting him. If one sees that his rebuke will have no effect, due to the great evil of the sinner, or if the sinner is intimidating and might kill him, he is exempt.

Poskim

Rema (OC 608:2): If one knows that his words will not be heeded, he should rebuke publicly only once. However, privately one should rebuke until the point that the transgressor is ready to hit or curse him.

Source (Gra DH Rak): In Shabbos (55a) R. Simon thought that he was exempt from rebuking because the rebuke needed to be in public (but he knew that privately one must rebuke repeatedly).

Taz (2): Tosfos (Bava Basra 60b DH Mutav) learns from Shabbos 55a that if one is unsure if his rebuke will be heeded, he must rebuke. Shabbos 148b teaches that one does not rebuke if he knows that it will not be heeded.

Magen Avraham (3): The Rema discusses when the transgressor is Shogeg. If he is Mezid, one is not punished like him, but if he does not rebuke, he transgresses a Mitzvas Aseh. One must rebuke him until he gets angry. Afterwards, one may not rebuke him - "don't rebuke a scoffer". For an Isur in private, one rebukes in private. For an Isur in public, one rebukes immediately, to avert Chilul Hash-m. Semak rules like the opinion that one must rebuke to the point of anger. Sefer Chasidim says that this applies to a friend. One should not rebuke one who will hate him and take vengeance.

Levushei Serad (5): The Rema wrote 'until the transgressor is ready to hit or curse him.' He was unsure which opinion we follow.

Machatzis ha'Shekel (DH Semak): I say that the Rema obligates rebuke until he is ready to hit him, and permits rebuking until he is ready to curse him. The dearness of Hash-m to us (we are willing to go beyond the letter of the law to protest for His honor) is no less than David's dearness to Yehonason.

R. Akiva Eiger: I am unsure if we say that it is better that people be Shogeg about less than a Shi'ur of Isur, for this is not explicit in the Torah.

Mishnah Berurah (7): Whenever there is danger, one need not protest.

Mishnah Berurah (9,11): We rebuke once, in case it will be heeded, or so the transgressor will not be able to defend himself. Some say that it suffices to protest until the transgressor gets angry. One may rely on this, for we may join the opinion that one need not protest when he knows that his words will not be heeded (Sha'ar ha'Tziyon 13).