This is very subtle and orally this is often ignored as you will use 'de' when there is a clash of vowels-
e.g. Il a continué d'amener les enfants à l'école. ( He carried on taking the children to school.)
Normally, you will use 'continuer de' when a habit cannot be broken :
Il continue de fumer malgré ses problèmes de santé. (He carries on smoking in spite of his health problems.)
Il continue de me harceler. (He carries on harassing me.)
Nous continuerons de servir les clients à table ici! (We'll continue table service in this establishment!)
Continuer à is used when an action has started which carries on in time:
Vous continuerez à lire cette lettre chez vous. (You'll continue reading this letter at home.)
Elle continue à être dans cette situation précaire. (She continues being in this delicate situation.)
This is the theory but you will hear it used more loosely in spoken French as the French often mistake one for the other.
Hope this helps!