If you are NOT getting notifications when in range of your WiFi network, it's possible that you are at the FRINGE of the coverage for your home network.
What happens is the phone get's JUST enough of a signal from the router to connect, but not enough for data to go through. The problem is, this confuses the phone. It thinks it's on WiFi and tries to route all data through that. But, it's too weak of a signal. Turn WiFi off and the data runs through the 3G/4G.
You don't WANT the WiFi to stay on all the time. If it did, your battery would last just a couple of hours, even without you using the phone. The 3 biggest battery killers on a phone are 1) The Display, 2) GPS, and 3) the WiFi radio.
Why do you think apps that turn your wifi on and off based on your location (Cell towers) are so popular?