Correct spelling, explanation: continuing can be used as a present continuous form or an adjective based on the verb continue. What is important is to remember that the ending e of continue is not here. This is because when there is a vowel clump ei, it’s just often better to simplify it. In the case of the ing suffix, usually, the ending e is dropped – and that’s exactly why we write continuing.

Definition of continuing:
1. constant, unstopping
We couldn’t host a barbecue because of continuing rain.
To be honest, I have no interest in continuing this pointless conversation.

Incorrect spelling, explanation: this version (continueing) may seem convincing, but remember that the language doesn’t like when there is too much of something – and -eing is not a usual ending in English. It’s unnecessary, and we read only the ing anyway. When you look at them, most verbs ending in e tend to lose it when the -ing suffix is added, and, continue is not an exception.