Sorry about the delay in getting back to you...
Demeurer as your rightly point out is an odd one as it doesn't really fit into the usual transitive and intransitive category of verbs which take avoir or être.
However, it does mean two different things and use both auxiliaries depending on its meaning.
Maybe a better way of thinking about it is :
When the verb demeurer describes a state it will take 'être' and when it describes an action, it will take 'avoir'.
However, this is just a particularity of that verb and there is no need to sweat over as you will often use other verbs to describe the same meaning, rester in the case of the state of being ,
Je suis demeuré/resté bouche bée = I stayed open mouthed
and vivre/habiter for the action,
Nous avons demeuré/vécu/habité rue St Martin pendant dix ans = We lived in Rue St Martin for ten years
You will find exceptions to all rules so this is very perceptive of you...'Vivre' is the same , intransitive but takes 'avoir'
J'ai vécu = I have lived
Hope this helps!
Not quite sure if I have understood your question but in the 'Grammaphile's Corner' section of the lesson, it is explained that when the verb demeurer is transitive ( meaning it has an object) it uses 'avoir' and intransitive ( without object) it uses 'être' which seems consistent to me.
Cecile's answer to Tim really helped me. I was also confused about the opposite behaviour of demeurer relative to other verbs that use etre or avoir depending on context. It would be very helpful if the lesson was altered to explicitly state that this verb is odd - all the other lessons state that if the verb is followed by a preposition (en, sur, dans, à etc) the auxillary is etre, but that is not the case for demeurer (e.g. Il a demeuré à Paris pendant quelques mois.). Explaining that it breaks the rules, and adding the core of Cecile's explanation would avert confusion. This comment from Cecile was particularly helpful: "When the verb demeurer describes a state it will take 'être' and when it describes an action, it will take 'avoir'."
The lesson is going to be adapted and your comments will be taken into consideration.
Thank you very much for pointing those out...
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