Luke 7:1 When Jesus had finished saying all this to the people who were listening, he entered Capernaum. 2 There a centurion’s servant, whom his master valued highly, was sick and about to die. 3 The centurion heard of Jesus and sent some elders of the Jews to him, asking him to come and heal his servant. 4 When they came to Jesus, they pleaded earnestly with him, “This man deserves to have you do this, 5 because he loves our nation and has built our synagogue.” 6 So Jesus went with them.

He was not far from the house when the centurion sent friends to say to him: “Lord, don’t trouble yourself, for I do not deserve to have you come under my roof. 7 That is why I did not even consider myself worthy to come to you
But say the word, and my servant will be healed. 8 For I myself am a man under authority, with soldiers under me. I tell this one, ‘Go,’ and he goes; and that one, ‘Come,’ and he comes. I say to my servant, ‘Do this,’ and he does it.”
When Jesus heard this, he was amazed at him, and turning to the crowd following him, he said, “I tell you, I have not found such great faith even in Israel.” 10 Then the men who had been sent returned to the house and found the servant well.
What was it about the centurion’s extraordinary faith that led the Lord Jesus to say, “I have not found such great faith even in Israel”?
It was clear that the centurion had faith that Jesus could heal his servant. But there were others mentioned in the gospels who had similar faith, for example, the woman with the bleeding in Mark 5.  She believed that if she could just touch Jesus’ clothes, she would be healed.
Mark 5:25  And a woman was there who had been subject to bleeding for twelve years. 26 She had suffered a great deal under the care of many doctors and had spent all she had, yet instead of getting better she grew worse. 27 When she heard about Jesus, she came up behind him in the crowd and touched his cloak, 28 because she thought, “If I just touch his clothes, I will be healed.” 29 Immediately her bleeding stopped and she felt in her body that she was freed from her suffering.
30 At once Jesus realized that power had gone out from him. He turned around in the crowd and asked, “Who touched my clothes?”
31 “You see the people crowding against you,” his disciples answered, “and yet you can ask, ‘Who touched me?’ ”
32 But Jesus kept looking around to see who had done it. 33 Then the woman, knowing what had happened to her, came and fell at his feet and, trembling with fear, told him the whole truth. 34 He said to her, “Daughter, your faith has healed you. Go in peace and be freed from your suffering.”
According to Jesus, the woman’s faith healed her. She had faith that when she touched Jesus’ clothes she would be healed, and it came to pass just as she believed. Similarly the centurion believed that if Jesus simply “said the word”, his servant would be healed. How was the faith of the centurion greater than the faith of the woman who had been healed of her bleeding?
The key to understanding this is found in verses 7 and 8:
7 … But say the word, and my servant will be healed.  8 For I myself am a man under authority, with soldiers under me. I tell this one, ‘Go,’ and he goes; and that one, ‘Come,’ and he comes. I say to my servant, ‘Do this,’ and he does it.”
It was only after hearing these words above that Jesus said, “I tell you, I have not found such great faith even in Israel.”  What did those words tell Jesus about the faith of the centurion?
Notice first that the centurion did not ask Jesus to pray for his servant, but rather simply to “say the word.” Say what word? To answer this, look at verse 8 where the centurion—a military officer—explains his clear understanding of authority over the soldiers under him. When he tells one of his men to go, he knows that he will of course obey. When he commands another soldier to come, he knows the man of course will obey and come immediately. When he orders his servant to do something, he of course knows that he will promptly get it done.
From his words we know that the centurion—being a military officer—has ABSOLUTELY NO DOUBT that when he issues commands, the men under his authority will in fact obey. In such a way he showed that he understood EXACTLY HOW Jesus could heal his servant. Since he knew that infirmities were under Jesus’ authority, he could simply say the word—issue a command at a distance from wherever he was at the time—and the infirmity would have to obey and leave the dying servant. Since authority is not at all affected by distance, Jesus could simply say to the servant something like “be healed” from wherever he happened to be and it would be done.
When Jesus issued the command to the infirmity/servant, he had no doubt that it would obey him—just like the centurion had no doubt that men under his authority would obey his commands.
The person who truly understands authority will have no doubt that those things under his authority must and will obey his commands. And Jesus equated this unusual understanding of authority with great faithThis is in fact the mountain-moving faith which Jesus taught Peter after the disciple saw the fig tree wither at Jesus’ command in Mark 11.

Mark 11:23  “Truly I tell you, if anyone says to this mountain, ‘Go, throw yourself into the sea,’ and does not doubt in their heart but believes that what they say will happen, it will be done for them. 
If we can understand the intimate relationship between mountain-moving faith and authority, we can be like the centurion whose faith amazed Jesus. If we truly understand the nature of the authority the Lord has given us, then we can in fact move the “mountains” which he has put under our authority.

Application to healing the sick in the context of preaching the gospel
New Testament Scripture teaches that disciples have been given power and authority over infirmities and demons (“mountains”) for the purpose of proclaiming the kingdom of God to the lost.
Luke 9:1  When Jesus had called the Twelve together, he gave them power and authority to drive out all demons and to cure diseases, 2 and he sent them out to proclaim the kingdom of God and to heal the sick.
Luke 10:1  After this the Lord appointed seventy-two others and sent them two by two ahead of him to every town and place where he was about to go. ….9 Heal the sick who are there and tell them, ‘The kingdom of God has come near to you.’ 
How do we exercise this authority effectively for confirming to the lost with miraculous signs that the gospel is in fact true?
We issue commands to infirmities and demons just as the centurion gave orders to his men and just as Jesus “said the word” to the dying servant. We command and rebuke diseases and demons with no doubt based on the fact that the Lord has placed them under our authority for the sake of the gospel.