A pure substance has fixed and exact melting point.
The identity of a pure solid can be deduced by its melting point. For example, pure steatic acid melts at exactly 70 degrees celsius. THe presence of any impurities will lower this melting point. Also, if impurities are present, it will melt over a range of temperatures, perhaps 68-70 degrees celsius, as it is a mixture.
A pure substance has a fixed and exact boiling point.
The identity of a pure liquid can be deduced by its boiling point. For example, pure ethanol boils at exactly 78 degrees celsius. The presence of impurities will raise the boiling point, and also causes the liquid to boil over a range of temperatures.
A pure substace shows only one spot on a chromatogram.
Chromatography can also be used to test for purify. It is especially useful with complicated chemicals like drugs, of food addictives which cannot be easily melted or distilled. For example, using chromatography, a public anlyst can identify the synthetic dyes many foods contain and check that they are harmless. Many green vegetables sold in markets have been treated with pesticides and herbicies. These can be harmful to the people. Chromatography can detect such cehmicals even down to very low concentrations.