Most service providers mix up the terms modem and router.
Most devices can do both, route and modem.
You will always need modem functionality, because it transforms the dsl or line signal to tcp (those nice lan connections) and back. You can test if this works if you can ping the usual website of your provider.
Routers are needed to “route” the network traffic to the correct destination. (I.E. From wireless to physical network). These functionalities are usually built in the modem.
In most cases traffic comes from dsl, goes to hardwired and gets routed to wireless… then all back again.
For more information contact your internet service provider.