I have Shimano TZ rear derailleur. when I'm shifting down (7--1) it works fine but I shift up it malfunctions. like it won't come down when I shift it to 2, then when I shift to 7 it jumps from 5 to 7. I tried fixing the limit screw recently. I also tried indexing using the barrel adjuster. but I can't fix it no way. please help.

Try cleaning it first. Can use a toothbrush with soap and water, gently rinse off. Then lube the pivots with something like Tri-Flow or any good lube. Is the cable going to the der. moving freely? Is the shifter moving freely-not "hanging up". If the cable and shifter is OK, and the der. has been cleaned and lubed, it may be time for a new der. The TZ is a "budget" der.- might want to go up a level or two. EDIT-one other thing-has the bike been dropped on the der.side? Could be out of alignment, but you'd probably want a shop to check that.

First thing I'd do is pull the cable out and lubricate with triflow. Then move onto derailleur adjustment.

+1 Having a derailleur work one way and not the other is a classic sign of excess friction in the system, often caused by the cable. Pay special attention to the loop of cable going into the rear derailleur as it tends to collect road grit and other gunk.

If it worked previously and now it doesn't...and nothing has changed, it's not an adjustment issues, it's more likely cable/housing or maybe a cable fraying in the shifter. Undo the cable from the derailleur and confirm the derailleur moves properly. If it does then you know you have a cable or housing problem. Move on from there. If you shifting has problems and nothing has happened to the bike (bent hanger, damaged derailleur) don't ever just start messing w/ limit screws.

Has your drop out where the RD is mounted been aligned? And could your RD have gotten bent?

Sounds like you are off a little. That doesn't sound right, lol...

+1
Failure to shift to the smaller cogs/higher gear is because the RDER spring can't pull the sticky cable back reliably through the housing.
Colder weather may exacerbate the problem.
Shifting to larger cogs is a brute force kind of thing, where cable friction is simply overwhelmed.