Try a Window Comparator circuit using a dual op amp...

Lower limit puts heater on, upper limit puts fan on, in between nothing on...

 

Try a Window Comparator circuit using a dual op amp...

Lower limit puts heater on, upper limit puts fan on, in between nothing on...

That won't meet his specs!

Should the fan be on if the temperature is 1°C above the target temperature (and within the ±2°C window)? Answer: Maybe. It depends on information that is not captured by a window comparator. He needs hysteresis!

 

Yup. Exactly like your home's HVAC.