Edit: In all honesty, captains on here, from a company and legal viewpoint, are you expected to just carry on if this above scenario occurs mid-flight
No you aren't expected to carry on, you are expected to have loaded enough fuel to ensure that at all times in flight you have fuel onboard that is;
- enough to a suitable airport
-10% of the above
-special fuel
- reserve fuel etc etc
A suitable airport is one that has ok weather. So when presented with fog, the crew have to look for a 'suitable airport' that they can make. In practice this means you are always aware of which airports you can make and what the weather is doing there, if options start getting slim you start paying more attention.
Most of the time any issues can be sorted out at the pre flight stage by adding more fuel, many states mandate carrying an alternate on every flight ( not Australia) this reduces risk and removes potential for poor judgement or interpretation of the forecasts.