Preimage of Subset under Mapping/Examples/Identity Function with Discontinuity

Example of Preimage of Subset under Mapping
Let $f: \R \to \R$ be the real function defined as:
 * $\forall x \in \R: \map f x = \begin {cases} x & : x \ne 0 \\ 1 & : x = 0 \end {cases}$

Let the following subsets of $\R$ be defined:

Then the preimages under $f$ of these sets is:
 * $f^{-1} \sqbrk A = \set {x \in \R: \map f x < 0} = \openint \gets 0$


 * $f^{-1} \sqbrk B = \set {x \in \R: \map f x > 0} = \openint 0 \to$


 * $f^{-1} \sqbrk C = \set {x \in \R: 0 < \map f x < 1} = \openint 0 1$


 * $f^{-1} \sqbrk D = \set {x \in \R: -1 < \map f x < 1} = \openint {-1} 0 \cup \openint 0 1$


 * $f^{-1} \sqbrk {E \cup F} = \openint {-2} {-1} \cup \openint {\dfrac 1 2} 2 \cup \set 0$