How to appear offline on Facebook?
This article shows how to appear offline on Facebook while using desktop and mobile app. If you want to be totally incognito then you have to put yourself offline from both. Remember that when you are on offline mode, you can't see your friends online status, but you can still receive messages normally.
1. When you are logged on Facebook on desktop, locate this little gear icon on bottom right. Click that and then Turn Off Chat.
2. You can select best option that suits for your needs.
3. Open your Facebook mobile messenger and click icons as showed in picture below.