Kibutsuji Muzan, Demon Slayer's King of Demons, has few weaknesses, but he still must hide from the sun. For the centuries he has been alive, Muzan's goal is to concur death and live eternally. Muzan has gained powers during this long journey: the demon curse or Muzan's curse.
A demon can only get this curse by ingesting Muzan's blood. It seems easy to avoid, but power is both intoxicating and frightening for demons. Many follow Muzan because he has great power; they don't want to risk their lives, so they follow his command. So, what is this curse that some of Demon Slayer's strongest demons are willingly taking on?
Curse of Kibutsuji
This curse besotted any who partake of Muzan's blood must be careful. The curse prevents Muzan's name from being spoken aloud, and any who do will meet a horrible demise. One demon, Susamaru, faced this reality when battling Tamayo. Susamaru accidentally shot Muzan's name aloud, attempting to defend his honor with her mind clouded. Even though Susamaru was defending Muzan, the curse still took effect, causing three demon arms to burst from her chest. Leaving Susamaru a puddle of blood and guts, waiting for the sun to kill her finally.
Based on what happened to Susamaru, it's clear that the curse has no mercy; no matter the reason or context, it will destroy the demon that says Muzan's name. Knowing this if a demon slayer was clever enough, they could get a demon under Muzan to say his name, forcing them to face the curse. Then again, Muzan might trigger the curse himself, preventing the demon from being manipulated by another. This exact scenario happens (in the manga) when Yushiro is controlling Nakime. To stop Nakime from moving the Mansion, Muzan activates the curse, stopping Nakime.
Based on these two instances, it's safe to say that Muzan uses this curse as a safety percussion. Having what can be considered a time bomb in all of his followers means he doesn't have to worry about a betrayal. The moment they decide to speak his name, he can destroy them, or if they are betraying him, he can activate the curse instantly. Muzan is a cautious man. Even though he is the most powerful demon, he still isn't willing to risk his life instead of implementing safety measures everywhere. This doesn't mean everyone has fallen to the demon curse.
Those Who Survived
The two known demons who broke free of Muzan's curse are Tamayo and Nezuko. Tamayo was able to break free of the condemnation when Yoriichi fought Muzan, nearly killing him in the process. During this weakened state, Yoriichi let Tamayo go, free her from Muzan, and free her from the curse. With her newfound freedom, Tamayo made it her life's mission to see the death of Muzan finally become a reality.
Nezuko, on the other hand, never was under Muzan's curse, it would seem. Instead, she used her will to protect her brother and other humans to fight off the curse. This is probably why Nezuko sleeps instead of eating humans. This also means her strength as a demon is based on her will to protect rather than feed. If only other demons had discovered this sooner, then more might have ended up like Nezuko.
These two demons might be the only ones who broke free of Muzan's curse, but they show others that his power isn't unbreakable. One day the demon slayers will stop Muzan and end his reign. Until then, slayers will have to find new ways to implement this curse, hopefully getting them to say his real name or even break free from the curse.