So there might be problem for you to appear offline in Facebook. Let me make you clear that you can appear offline to anyone you like in Facebook or you can appear offline to everyone in Facebook. This is quite simple. May be you have known how to do this in previous Facebook meaning that before Facebook changed its look and you might find it difficult to appear offline in Facebook now. If this is the problem then let me make you clear that this is very easy.
To do this, just below the friend lists for chat, you can see small icon of settings. Click there and click on Available for chat. If this is checked right then you are available to chat or not. If you want to appear offline to specific person only then you will need to create a group and then add that people in group and click on Limit Availability. Then you can choose by yourself. This is real easy.
Click here and it will be done automatically