As was suggested above, detach the cable from the derailleur (i.e., loosen the bolt pinching the cable in place) and move the derailleur in and out by hand. If the derailleur springs back freely to its resting position, the problem is friction in the cable housing.

If you're reluctant to detach the cable from the derailleur for some reason, here's what you can do.

(i) Lift the rear wheel off the ground and then pedal and shift until the chain is on the biggest cog.
(ii) Leave the chain on the biggest cog and drop the rear wheel onto the ground.
(iii) Return the shifter to its resting position. (If it's an indexed shifter, you should pull down on the shift cable as you're moving the shifter to its resting position.)
(iv) Move the derailleur in and out by hand. If the derailleur springs back freely to its resting position, the problem is friction in the cable housing.

That's how you check the rear derailleur; checking the front derailleur would work the same way.

All of us who have run service departments in bike shops believe that the chances of it being anything other than a cable housing problem are remote.