Is salary a discrete or continuous variable?

Continuous

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Actually, it is discrete. In most countries that I can think of the currency has a major unit that is divided into 100 smaller units: dollars + cents, pounds + pence, Euros + cents, Rupees + paise etc.

You cannot have anyone earning between two consecutive numbers of the smaller unit. For example, between 1234.56 and 1234.57 units of currency.