Why does my Facebook chat show up as online for my friend?

[dollydaydream4jc]

I haven't knowingly listed myself as "online" on Facebook chat for years (except for recently in attempts to figure this out). The reason, of course, is that I was sick of people constantly messaging me just because I was there. Recently, a friend has started this practice with me whenever I'm online, except I wasn't listed as online as far as I knew, so at first I thought he was just being observant and seeing posts from me in his newsfeed, but then I realized he was often messaging me before I could even post anything, so I asked him, and he said that I sporadically show up as online, like the green circle blinks on and off or something. I sent something to Facebook's tech support, but they haven't gotten back to me, and it is obvious the issue still isn't fixed because this friend still messages me immediately and without fail every time I log on, despite the fact that my chat still says I am offline.
What can I do? Do I have to delete all my applications? Is there just some setting I'm not aware of? I just can't keep devoting my extra time to making polite conversation with someone who wants to talk to me more than my own mother and my best friend combined. Once in a while is fine, but...
I doubt it's important, but just in case, I'm using Firefox with Windows 7.

[Ony]

it happens when you keep turn on chat. remember to log out from chat.