First, we have to believe and learn the doctrine of the Church regarding infallibility, not only by the pope but of the entire church. The first Vatican Council precisely defined what ex-cathedra or infallibility decisions mean. In teaching infallibly, the pope has the same extent as the Church.
What does teaching infallibly mean? It means that the Pope or the Church has the assistance of the Holy Spirit when proposing definitively (called an ex-cathedra) a certain truth that God reveals to the entire church, the church cannot commit an error or teach heresy.
By catholic faith, it is impossible for the Catholic faith that the Pope would teach infallibly or in ex-cathedra an error or heresy and it has not happened and it will never happen in the 2000 years of history of the Church. The condition is that the Pope has to declare that he is teaching definitely, forcing all the members of the Church to accept these teachings as divinely revealed. These truths should be considered divine revelation and are contained in the deposit of faith. If one of these conditions is not met, then the Pope is not speaking ex-cathedra and is not assisted by the Holy Spirit. Therefore, when a Pope, a Bishop’s conference, or a council is teaching not definitely, there is no assistance from the Holy Spirit to keep them from infallibility,
There have been cases where a Pope or an Assembly of Bishops has spoken an error because they have not spoken definitely. Pope Francis especially is against speaking definitely, he is using a language that is ambiguous and his doctrines are developing therefore he has no intention of proposing something definite.
So, in these cases when he teaches an error or something ambiguous, a catholic does not have the obligation to follow him. Bishops and Lay faithful must respectfully admonish the Pope and correct some ambiguous teachings. This is not contrary to the Catholic Faith, this is what St. Paul did to St. Peter in Antioch, and in other instances in the history of the Church such as St. Catherine of Sienna, who also admonished the Pope during her time.
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