if you block someone from facebook chat when they search your name will it still show you on facebook mobile?

[J]

i want to block chat from someone on Facebook, if i do this will it still show up that you can be messaged to your phone or will your name not appear to the person at all? even if they search for you
but if they search my name in the chat window will it appear at all? or will it appear with the little phone symbol next to it all the time?

[Run4Fashion]

The message will go to "Messages," and it may go to your phone if you have enabled notifications of messages on your phone, either from Facebook itself or text messaging