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According to my textbook:

A function which has either $\mathbb R$ or one of its subsets as its range is called a real valued function. Further, if its domain is also either $\mathbb R$ or a subset of $\mathbb R$, it is called a real function.

As there are 2 definitions here, is there a difference between "real functions" and "real-valued functions"?

MathWorld says that a real function is also called a real-valued function.

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  • $\begingroup$ Each book should define the terms it will use. Especially if the meanings differ or are not well-known. $\endgroup$ – GEdgar Jan 1 '16 at 14:17
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According to these definitions, any function $\mathbb C\to\mathbb R$ (for example, $z\mapsto |z|$) will be a real-valued function but not a real function.

As your research shows, this usage is not universal -- there can't be much disagreement about what a "real-valued function" is, but how the words "real function" are used can depend on the author and field.

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  • $\begingroup$ Yes it's real valued but not a real function... $\endgroup$ – neelkanth Jan 1 '16 at 13:59

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