Boyfriend broke up with me but still texts me everyday?

We've only been together for a week, he told me he cheated on his ex before but he is not like that anymore (explained to me why he told me that so that if later I found out how he was before I would think he had been lying to me). He said he is serious wanting to be with me...I have some trust issues so I'm sensitive to things, he got tired of it and said we should be friends again for now until I know him better as a person then we could start again then. I don't know if he really meant what he said or he is just letting me off easily...since we broke up I thought I wasn't going to hear from him but everyday I'll get a text from him about an hr into my work day, he'll ask me how's work, how I'm doing, if work is busy, where I'm going for lunch, etc. then in between him going to work and I'm getting off work we don't really talk but then later after a few hrs he'll text me again what I'm doing, if I'm home yet...I don't reply him right away maybe about 10-20min later..and I try to keep it as short as possible not asking him anything. I'm not sure if I should be even replying him? I'm afraid if I don't reply him a few days he's going to back off and not talk to me at all and think I don't care about him anymore..I'm so clueless what to do, I've been working so hard to not be needy and desperate because I want to be with him. He's been the one initiating the contact for now by text only though..late at night the texting doesn't continue, then I go to work and same thing all over; that's how the day repeats; so guys and gals what's the deal? Should I just keep replying his texts or ignore them or what? I don't want to play games, definitely not sure if he is playing any games. Do you think he really wants me to know how he is as a person, so that I can trust him then we can start the relationship over?
Boyfriend broke up with me but still texts me everyday?
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