Incorrect spelling, explanation: in cases when the suffix starts with a vowel, and the word also ends with a vowel, there is sometimes this problem of vowel clusters; it seems like it makes sense, after all, its base word (continue) and the proper suffix, but it’s worth to remember that language (or rather its users) like to make things as easy as possible. This is why in cases like continueing, the e has to go to make place for a simpler and more intuitive continuing.

Correct spelling, explanation: continuing is an adjective based on the word continue. In the process of adding the -ing suffix, the ending e of continue is exchanged for the -ing. These kinds of changes happen often, whenever there are two vowels put together and the words becomes difficult to spell with too many vowels.

Definition of continuing:
Adjective
1. Not stopping, ongoing
The continuing demand made us bring our special burger back.
They are still continuing to push their business forward.