Unfortunately, this means that is not a proper function but is instead a "multi-valued function": for example, any positive real number has argument 0, but also has argument . This means that the argument may be best considered as an equivalence class . The advantages of this are several: most importantly, they make into a continuous function. They also make some properties of the argument "look nicer." For example, under this interpretation, we can write . The other common solution is to restrict the range of to some interval, usually or . This forces us to state this equality modulo .
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