Why is it called 'discrete' mathematics?

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I understand why you would refer to mathematics which concerns itself with all of the numbers on the number line as 'continuous' but why would you refer to countable or finite mathematics as 'discrete'? Like it's the mathematics no one talks about? Where does this labeling come from?

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As the online dictionary at m-w.com says, "discrete" means anything constituting a separate entity or anything consisting of distinct or unconnected elements.