How can faith in Jesus Christ come from anything that is me acting apart from God?
You answered your own question. Your faith is you choosing to believe in something.
If you believe in God it is because you decided by the inspiration of God to believe in Him.
I have absolutely no control over the fact that I know Jesus Christ is God, my God. So how is this faith credited to me as though it is something I am generating from myself?
That is actually a good thing you feel that way. But don't let that feeling make you think that you did not chose to have faith. Everyone should feel compelled because the evidence is so obvious, however Jesus was never afraid to credit people the power of THEIR own faith in Him:
Matthew 9:22 Jesus turned and saw her. "Take heart, daughter," he said,
"your faith has healed you." And the woman was healed from that moment.
Mat 9:29 29Then he touched their eyes and said, "
According to your faith will it be done to you";
Luke 17:19 Then he said to him, "Rise and go;
your faith has made you well."
It is part of my life. I cannot stop believing in Jesus anymore than I can stop believing in me!
good.
Why do i hear "ye must continue to believe, ye must continue to believe"?How can I not?
It's simple, because it's possible for someone to not STOP believing. If you accept all the teaching in scripture you will accept the warnings as well. It's not that people stop believing that Jesus is real and that he can save or that God exist, no not at all.
In some cases people kept ignoring the spirit and started deliberately sinning while knowing Jesus such that nothing else can save them (meaning there is no sacrifice for sins is left, but only a fearful expectation that will consume the enemies of God) Heb 10:26
In other cases some people go arrogant
Rom 11: 17 If some of the branches have been broken off, and you, though a wild olive shoot, have been grafted in among the others and now share in the nourishing sap from the olive root, 18 do not boast over those branches. If you do, consider this: You do not support the root, but the root supports you. 19 You will say then, "Branches were broken off so that I could be grafted in." 20 Granted. But they were broken off because of unbelief, and you stand by faith.
Do not be arrogant, but be afraid. 21 For if God did not spare the natural branches,
he will not spare you either.
If you are able to accept the warnings as they truly are it should give you Godly fear which is not of yourself.
Jesus is no different in John 15: (vine and branches). He expects fruit from those connected to the vine, but if no fruit GOD HIMSELF WILL CUT Loose the branch.
Well Jesus knew hearts will associate expectation with harshness or lack of grace, so he gave another parable for the believers
Matthew 25:
The Parable of the Talents
14"Again, it will be like a man going on a journey, who called his servants and entrusted his property to them. 15 To one he gave five talents of money, to another two talents, and to another one talent, each according to his ability. Then he went on his journey. 16 The man who had received the five talents went at once and put his money to work and gained five more. 17 So also, the one with the two talents gained two more.
18But the man who had received the one talent went off, dug a hole in the ground and hid his master's money.
Notice two when straight to work believing the master and showing trustworthiness, BUT the other chickened out and did not even attempt to apply himself (remember his still a servant)
This is that servants response:
24 "Then the man who had received the one talent came. 'Master,' he said,
'I knew that you are a hard man , harvesting where you have not sown and gathering where you have not scattered seed. 25 So I was afraid and went out and hid your talent in the ground. See, here is what belongs to you.'
Now this guy produced nothing because he misunderstood the master calling him a hard man because he gave the servant something to do
26 "His master replied,
'You wicked, lazy servant! 
So you knew that I harvest where I have not sown and gather where I have not scattered seed? 27Well then, you should have put my money on deposit with the bankers, so that when I returned I would have received it back with interest.
God can use the least to the greatest of us, but he will not tolerate lazy excuse making servants
Well, so much for the "God knows that I can do nothing" mentality
End result of not someone who didn't believe, but some one who believe, who didn't care enough to do something with it:
(faith without deeds)
28" 'Take the talent from him and give it to the one who has the ten talents. 29 For everyone who has will be given more, and he will have an abundance. Whoever does not have, even what he has will be taken from him. 30 And throw that
worthless servant outside, into the
darkness , where there will be weeping and gnashing of teeth.'
Now don't go applying this to unbelievers again, because in order to be thrown into darkness, you would have 1st needed to be in the Light
Think about it.