[A relevant response from Chris Perkins](https://www.sageadvice.eu/2015/07/16/what-happens-if-a-magic-item-is-broken/) (story designer for D&D):

> *What happens if an Eversmoking Bottle is broken? Does the smoke cloud expand on and on forever?*
>
> If a magic item breaks, it ceases to function. The eversmoking bottle would stop producing smoke.

That said, by RAW, I don't think there are any rules about whether any break in a magic item causes it to lose its magic, other than the descriptions of specific magic items that state this property of those particular items (e.g. the Staff of Power or Staff of the Magi's Retributive Strike ability). Ostensibly, if you totally broke a magic item, it would not longer be able to function as normal.