I promised my mother I would continue (attending something or doing a particular task), but I broke the promise. What is the ruling?

If "I will continue" refers to continuing in a job, then it is an obligation according to the contract, and breaching it is a sin. As for fulfilling a promise in general, it is recommended by the majority of scholars, and breaking it is mildly disliked (makruh tanzihi), unless there is a need for it, in which case it becomes obligatory. If the promise concerns an obligatory matter, its obligation is emphasized, especially if it is made to one's mother, due to the aspect of filial piety involved. It is advisable to be diligent in fulfilling promises.