Functor - what power do they have over other classes that didn't implement operator ( )?

For one, they have a unified signature that can be used e.g. By algorithms that only need to know that they have to call something.

Functors do not have any special properties over classes, but they can be used as functions with higher capabilities. A function usually cannot have state, and if it has (by using a static variable) the state is shared among all function calls.

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