Is it possible to continue using an to external monitor when closing the lid on a Macbook Air? I have an application I use on an external monitor with an external USB device.

Is there a way to close the lid and continue using the app on the external monitor?

I turned off the sleep upon close function, but the external monitor still turns off.

OSX 10.9.5
Yes, its called "closed clamshell" mode. In short, you'll need to attach your MBA to a mouse, a keyboard and power:

https://support.apple.com/HT201834