I know that through faith alone I am saved. But is repentance a requirement for that salvation as a part of my faith? Is repentance a single act, or should it be continuous?
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Short Answer, YES; Long Answer, YES YES
First we need to define what repentance is.
The word in Greek is Metanoeo which means: To change ones mind. The word repent in Hebrew is Nacham which means to: to be sorry, be moved to pity, have compassion.
Now that we understand the word's definition we need to understand the context that the word is used in. Here in the old testament is one verse, of many, that defines repentance.
So you can see that repentance is a mindful state of abhorrence to the things that you have done. The bible says that we need to prove this mindful state by our actions. Have a look:
Repentance is: Having a changed mind that turns you from the things that you were once doing in order that you may do the opposite.
What does God say about Repentance?
We are commanded to repent of our sins.
What does Jesus say about Repentance?
Here we see Jesus talking about the final judgement, he's giving us insight into what He will say and do on that day.
We know that sin is lawlessness. Christ is literally saying that if you practice sin He will have nothing to do with you, you did not know Him and He did not know you. In this next verse you see Jesus commanding us to repent, He does this constantly throughout his testimony on this earth.
It's interesting what Luke says here, a baptism of repentance FOR the forgiveness of sins. If we put this altogether with what else we have learned so far, we see that if we turn from our sins then God will be faithful and forgive us for those sins.
So if this is true, then the opposite should also be true. If we continue to sin then we will go to Hell? Let's see if what the Bible says.
Finally God says that He stands FIRM on this very statement. He will not change His mind.
Assurance of salvation as it relates to repentance
If you continue to sin then the truth [Holy spirit, God, Jesus] is not in you.
If you say that you know God, but you continue to sin, not loving as you were commanded, then you are a liar and [Holy spirit, God, Jesus] is not in you.
Repentance is a struggle against the desires of your flesh
You will not be perfect right away, you will go through a lot of discipline and correction from God, so that you may produce more and more fruit throughout the days of your life. Persevere in your repentance.
Yes, it's a requirement... or better, a precondition. In order to be saved, you have to accept Jesus sacrifice. But for this, you have to realize that this sacrifice was necessary. And for that, you have to realize... your sin.
Now... you could accept your sin ("Yup, bad stuff")... without leaving it. Makes sense? Don't think so, but people may be wanting to see it that way. Here, the gospel (and that means, Jesus) is quite clear: you cannot accept the gift of life but keep to your old ways. Doesn't work that way (and it should be quite obvious why... would you forgive someone if you knew he'd keep on doing what you just forgave him?)
As to the second part of your question: it is both. It is a single act, when you accept Jesus as your Savior and Lord. And it's a sort-of ongoing process... because, being imperfect sinners, we always fall. So, we have to pick ourselves up and continue (stumbling) on. That means repentance every time we fall.
But beware: do not dwell on sin. No matter how big and nasty it was, it is already forgiven. Jesus died once, he doesn't die again every time you need forgiveness. The longer you dwell on it, the longer the enemy can keep you away from God. Doing whatever it was is bad enough, but staying away from God because of it makes it worse. He wants us close, and be close to us, always.
Standard Disclaimer: I do not speak for any tradition, denomination, or what-have-you. Authority? Ask God to show you if what I say is true. Then, either he confirms or he doesn't. That's all. (Dang, why doesn't this thing not have support for a signature or similar...)
Kudos to @JonathonByrd for his answer! I would just like to add a verse that I recently stumbled on in Luke:
The moment someone truly accepts Jesus, they have remorse for their sins, and they are saved. Once we have our salvation, we cannot lose it (John 10:28-29). However, you will continue to sin, so therefore you should continuously repent for those sins. Not necessarily for salvation, but to have a better relationship with the Lord.
Also, when we deliberately sin, and don't repent to God (maybe because we feel ashamed and/or don't really feel sorry because we keep doing it), we become more numb to the sin and it brings less conviction and takes us farther from God each time we do it.
Repentance from sin is required for salvation. We know this since the demons "believe". They will not be saved since God has prophesied their destiny in Revelation and elsewhere. Repentance from sin does not mean sinless perfection, it means a true, heartfelt decision that results in a change of behavior.
However, faith and repentance go hand in hand. If a person has no faith, why would he need or want to repent from sin? God's kindness in the forgiveness of sins leads us to repentance from sin. When we are saved, the Spirit of God gives us the power and desire to do so.