engine revving between shifts

does anyone know what the cause of this is, and how to fix it?
rsvp thank you

Mine did this when the computer was adjusting to the cold air intake it went away with time though.
Try unplugging the negative of the battery for 15 mins or so to reset the computer

explain how you shift?

the dbw is slightly delayed if your pushing the clutch in before taking your foot completely from the gas, the delay will increase rpm (especially with no load).

does the car do it if you "practice" in neutral? like the same revving of the engine?

There is no fix. The dbw system is slow to react and taking the load of the engine, i.e. pushing the clutch in, will cause the revs to go up because, although you've taken your foot off the throttle, the throttle plate is slowly returning to idle position. It also happens in the Corolla that I drive.

I wondered if the car was programmed to do this to avoid a temporary rich condition, which would shorten the life of the oxy sensor and cat.

ie; your on the gas one minute, then suddenly unloading the engine, by pushing in the clutch, while the injectors would still be spraying fuel since your technically still on the gas.

I do this for a smoother shift.. mine seems to hold the revs, when I'm in traffic that starts moving in an unpredictable stop and go situation.

You can shift into a higher gear like 3rd or 4th and let the clutch out for a sec to bring the revs down if it's really going nuts..