A matchstick tip conatins chemicals like Red Phosphorus, Sulfur, Potassium Chlorate, and sometimes Ammonium Phosphate and Paraffin Wax. 

Red Phosphorus acts as an igniter. When the matchstick is struck, the friction produces heat energy and red phosphorus causes combustion.

Sulfur acts as a fuel and keeps the flame up for a specific time.

The Potassium Chlorate acts as an oxidizer. The heat generated from the friction breaks down Potassium Chlorate and in this process, Oxygen is release

This Oxygen, along with Sulfur keeps the flame up.

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