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Can you make fist?
Are you taking any medicine now?
Did the pain start 5 days back or recently?
The tendons of the finger are running through the palm.
In case of deep cut there can be injury to them.
As you can make fist, there is not any serious injury.
Still, I would suggest to get an examination by doctor.
There can be an inflammation or deep seated infection,though the skin is healed.
You can take antiinflammatory analgesics like motrin or naproxen, if there is no other contraindication.
Swelling may not be present initially if it is deep seated.
Yes, it will help to reduce the inflammation.
Yes, it would not fight infection.
Initially, it may present similar. But with infection the symptoms would progressively get worse without treatment. There will be increasing pain, development of swelling, rise of temperature.