I have a braeburn 5000 thermostat. I have programmed it, but when it reaches the given temperature say 65, it doesn't cut off and start up when it goes below the set temperature. The furnace continues to run until I switch off the heat. I replaced the batteries and put in a new air filter...

The thermostat may be wired and/or set-up incorrectly, or the heating system may be malfunctioning. If you have thermostat installation literature, verify correct wiring & setup options.