Vatican I declared,
“‘For the fathers of the Fourth Council of Constantinople, following closely in the footsteps of their predecessors, made this solemn profession: ‘The first condition of salvation is to keep the norm of the true Faith. For it is impossible that the words of our Lord Jesus Christ Who said, ‘Thou art Peter, and upon this rock I will build my Church’ (Matt. 16:18), should not be verified. And their truth has been proved by the course of history, for in the Apostolic See the Catholic religion has always been kept unsullied, and its teaching kept holy.’ …for they fully realized that this See of St. Peter always remains unimpaired by any error, according to the divine promise of our Lord and Savior made to the prince of his disciples, ‘I have prayed for thee, that thy faith may not fail; and do thou, when once thou has turned again, strengthen thy brethren’ (Luke 22:32)
“‘So, this gift of truth and a never failing faith was divinely conferred upon Peter and his successors in this chair, that they might administer their high duty for the salvation of all; that the entire flock of Christ, turned away by them from the poisonous food of error, might be nourished on the sustenance of heavenly doctrine, that with the occasion of schism removed the whole Church might be saved as one, and relying on her foundation might stay firm against the gates of hell.”
Questions:
1. Has Francis kept the Catholic religion unsullied and teaching holy?
2. Has Francis remained unimpaired by any error?
3. Does Francis have unfailing faith from Christ’s prayer?
4. Does Francis strengthen his brethren with the Catholic Faith?
5. Has Francis turned the poisonous food of error away from the flock of Christ?
6. Has Francis nourished the Catholic flock with heavenly doctrine?
7. Has Francis removed all occasion of schism that the Church might be saved as one?
8. Does Francis stay firm against the gates of hell?
Speray's Catholicism in a Nutshell 
You can be sure certain people won’t deal directly with these questions. Or they’ll quote selectively from Francis.
I have previously posted on another blog the following quote from a source that received the imprimatur during the reign of Pope St. Pius X – the Catholic Encyclopedia. (The Q & A format and emphasis are mine.)
Question: Is a formal sentence of deposition required to deprive a heretical pope of his office?
Answer: No (according to many theologians).
Source quote:
“A similar exceptional situation might arise were a pope to become a public heretic, i.e., were he publicly and officially to teach some doctrine clearly opposed to what has been defined as de fide catholicâ. But in this case many theologians hold that no formal sentence of deposition would be required, as, by becoming a public heretic, the pope would ipso facto cease to be pope. This, however, is a hypothetical case which has never actually occurred; even the case of Honorius, were it proved that he taught the Monothelite heresy, would not be a case in point.”
A standard “R&R” response to this type of statement is that “the Church has not decided this issue infallibly”. The fact, however, that a truth has not been defined infallibly does not stop it being true or prevent us from relying on it. For example, it is not infallible that 2 + 2 = 4.
Come to that, the Church has never defined infallibly that Jorge Bergoglio is Pope.