Mormons claim that the authority of the Church was lost with the death of the Apostles, but what they don't realize is that the Apostolic Fathers themselves testified that they taught and carried the same Gospel.
St. Ireaneus writes:
"Polycarp, a man who had been instructed by the apostles, and had familiar intercourse with many that had seen Christ, and had also been appointed bishop by the apostles in Asia, in the church at Smyrna, whom we also have seen in our youth, for he lived a long time, and to a very advanced age, when, after a glorious and most distinguished martyrdom, he departed this life. He always taught what he had learned from the apostles, what the church had handed down, and what is the only true doctrine." (Irenaeus, book 3 against heresies, also Eusebius Book 4, chapter 14)
In another place, Ireanus describes the intimacy of John and Polycarp. The historian Pressense writes:
"Like St. Peter and St. Paul, Ignatius came to his end obscurely. Nothing is more remote from the melodramatic than the death of the saints. He left behind him, in Asia Minor, a young man, raised, perhaps by John himself, to the office of elder in the Church of Smyrna, and destined to exercise a great influence over the Christians in those countries. This young man was Polycarp. Ignatius had already noted in him remarkable steadfastness in the faith. He was planted upon the rock of apostolic teaching. The Church which he governed was one of the most flourishing in Asia Minor, and is exhibited to us in the Revelation as displaying courageous fidelity under persecution. Polycarp had been the immediate disciple of St. John, and ever cherished his sacred memory. It was the constant theme of his conversation and preaching. Irenaeus, who was the disciple of Polycarp, writes: "I could point out the spot where the blessed Polycarp sat to teach. I could describe his gait, his countenance, all his habits, even the clothes he was accustomed to wear. I could repeat the discourses which he delivered to the people, and recall all that he said of his intimacy with St. John, and the narratives he used to relate about those who had seen the Lord upon earth. His memory was constantly dwelling on that which they had told him of the words, the miracles, the doctrine of Christ. This valuable testimony shows how eminently qualified was Polycarp, for effecting the transition from the apostolic to the following age. He delighted to be the docile, almost passive echo of the apostles. It is not surprising, therefore, that he should not have displayed much originality, though commanding such universal respect. He was the living tradition of the Church." (Pressense, The Martyrs And Apologists, Hodder And Stoughton, London, 1870, 246-247. See also Eusebius Book 5, Chapter 20.)
Eusebius, careful to separate fact from fiction, tells us more of Polycarp and his studies under St. John. He also points out that St. Papias, Bishop of Hierapolis, had not actually know the apostles personally, though had many conversations with those Presbyters who were their immediate disciples. (Some scholars refute this, saying that some of the "Presbyters" may have been other apostles such as Andrew.)
"About this time, Polycarp flourished in Asia, an intimate disciple of the apostles who received the episcopate of the church at Smyrna, at the hands of the eyewitnesses and servants of the Lord. At this time, also, Papias was well-known as bishop of the church at Hierapolis, a man well-skilled in all manner of learning and well-acquainted with the Scriptures. Ignatius, also, who is celebrated by many even to this day as the successor of Peter at Antioch, was the second who obtained the episcopal office there....Of those who flourished in these times, Quadratus is said to have been distinguished for his prophetic gifts. There were many others also noted in these times who held the first rank in the apostolic succession. These, as the holy disciples of such men, built up the churches where foundations had been previously laid in every place by the apostles....As it is impossible for us to give the numbers of the individuals who became pastors or evangelists during the first immediate succession from the apostles in the churches throughout the world, we have only recorded those by name in our history, of whom we have received the traditional account as it was delivered in the various comments on the apostolic doctrine still extant....There are said to be five books of Papias, which bear the title Interpretation of Our Lords Declarations. Iranaeus also made mention of these as the only works written by him, in the following terms: "These things are attested by Papias, who was Johns hearer and the associate of Polycarp, an ancient writer, who mentions them in the fourth book of his works. For he has written a work in five books. So far Irenaeus. But Papias himself, in the preface to his discourses, by no means asserted that he was a hearer and an eyewitness of the holy apostles but informed us that he received the doctrines of faith from their intimate friends, which he stated in the following words: I shall not regret to subjoin to my interpretations, also for your benefit, whatsoever I have at any time accurately ascertained and treasured up in my memory, as I have received it from the elders, and have recorded it in order to give additional confirmation to the truth, by my testimony. For I have never, like many, delighted to hear those that tell many things, but those that teach the truth, neither those that record foreign precepts, but those that are given from the Lord, to our faith, and that came from the truth itself. But if I met with anyone who had been a follower of the elders anywhere, I made it a point to inquire what were the declarations of the elders. What was said by Andrew, Peter or Philip. What by Thomas, James, John, Matthew, or any other of the disciples of the Lord; for I do not think that I derived so much benefit from books as from the living voice of those that are still surviving." (Eusebius, Ecclesiastical History, Book 3, Chapters 36-37, 39.)
We also possess records of questionable authorship that testify of Apostolic authority:
"In the third year of Claudius Caesar, Simon Cephas departed from Antioch to go to Rome. And as he passed on he preached in the divers countries the word of our Lord. And, when he had nearly arrived there, many had heard of it, and went out to meet him, and the whole church received him with great joy....And after these years Nero Caesar seized him and shut him up in prison. And he knew that he would crucify him; so he called Linus, the deacon, and made him bishop in his stead in Rome." (The Teaching of Simon Cephas In The City Of Rome, from The Memoirs Of Edessa And Ancient Syriac Documents, Ante-Nicean Fathers 8:673-675.)
Though this record is today considered apocryphal, it nevertheless demonstrates the common understanding that authority was passed on, and has been preserved in the Catholic Church.
There is also another testimony of authority not yet discussed. These records are rarely discussed, but in themselves point towards a faithful church blossoming in the 2nd and 3rd Centuries. Besides those things just described, we have the testimony of the Church that various persons, intimately acquainted with Christ and His Apostles, were still living! According to the early Church Fathers, the saints resurrected with our Lord, continued on with the Church for quite some time (see St. Matthew 27:52-53):
"St John the Evangelist had a famous disciple, St. Polycarp, Bishop of Smyrna (died in 156). St. Polycarp, in turn, had a celebrated disciple, the great St. Irenaeus ( c. 130-220) who became Bishop of Lyons. Irenaeus know the early Church very well, east and west; as a famous apologist for the Church, he wrote five influential volumes entitled Against the Heresies. In that work Irenaeus says, "Some persons that were dead have been raised again and have continued among us many years." (Father Albert J. Hebert, Raised From The Dead, 33.)
"Quadratus addressed a discourse to him [Aelius Hadrian] as an apology for the religion that we profess because certain malicious persons attempted to harass our brethren. The work is still in the hands of some of the brethren, as also in our own, from which anyone may see evident proof, both of the understanding of the man and of his apostolic faith. This writer showed the antiquity of the age in which he lived in these passages: The deeds of our Savior, said he, were always before you, for they were true miracles; those that were healed, those that were raised from the dead, who were seen, not only when healed and when raised, but were always present. They remained living a long time, not only while our Lord was on earth, but likewise when he had left the earth. So that some of them have also lived to our own times. Such was Quadratus. (Eusebius, Book 4, Chapter 3.)
It seems rather incredible that those who had been raised from the dead, would then teach false doctrines, or allow those around them to preach that which was never taught. In this, if nothing else, we can be assured that the Church preserved and guarded those teachings, called the Deposit of Faith, well into the first few centuries. Proving that it continues to this day is question best left for another discussion.