Mapping

Consider the mapping \(F : \mathbb{R} \to \mathbb{Z} \times [0,1)\) defined as

\[F(x) = \left( \lfloor x \rfloor, \{x\} \right)\]

for all \(x \in \mathbb{R}\). Is this mapping injective? Is this mapping surjective?

Notations:

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