What a wireless router really is

The name lets suggest the following functionality:
Access point: provides a connection for wireless devices to the LAN
Routing: realizes the routing between the LAN subnets and between LAN and WAN

But most of them do actual way more than just that:
Switch: provides multiple connection ports for wired devices to the LAN
DHCP-Server: automatically assigns IP addresses to the LAN components
DNS-Server: resolves domain names to IP address (if unknown, forward to ISP)
Gateway: is an exit/entry point for the communication between LAN and WAN
NAT: translates all private IP’s (LAN) to a public IP (WAN) and vice versa
Modem: de-/modulates the signals for the traffic between LAN and WAN
Firewall: allows traffic from the WAN only if it was initiated from the LAN