Mazda released a bulletin 09/06 pertaining to this exact same issue, They had defective driver display units on many cars and had to issue and updated dispaly unit to correct the problem, Different symptoms can occur so to answer you on the problem, you need a new display unit, Now, I am not in parts but did a few quick checks and found some even on Ebay for under $89,The new part number that corrects the problme is also listed below, Beloe is also a copy of the bulletin Mazda issued, BTW: this is NOT a recall, it is a service bulletin for a part found to be defective
Here is new part number : GR1B-66-EVO
2006 MAZDA6 AND MAZDASPEED6 - INFORMATION DISPLAY DOES NOT ILLUMINATE/CLOCK RUNS FAST IN COLD OUTSIDE TEMPERATURES
2006 Mazda6 vehicles produced before January 3, 2006
2006 MAZDASPEED6 vehicles with VINs lower than JM1GG****6*104319
In sub temperature, some vehicles may experience no illumination from the information display when the ignition is turned on. Also, the clock runs too fast and may gain more than 20 minutes per week.