Do you really mean "pout"? That is a noun describing a facial expression. It comes from the verb "to pout", which means to mope or to sulk. So a "pout" is like when you have a sad look on your face, maybe because you didn't get your way or something. If that is what you mean, then you would say: I continue to pout. or I continue to keep the pout on my face. (although this is really awkward to say) I continue to keep pouting. I keep pouting. Maybe you mean "to keep out"? To stay out of something.

omit the 'keep', and then it is correct. You can either say, I continue to pout or I keep pouting but the construction "continue to keep pout" is not correct. Is like saying I continue to keep eat. This makes no sense, whatsoever. But saying, I continue to eat, or I keep eating, makes sense.

Keep out, or continue to pout? What are you trying to say Andre? The answer will change depending on your intended meaning.