Define a bitstring$-$to$-$bitstring function Fun via the following recursive rules:

Fun$(\backslash$lambda $)=\backslash$lambda $.$

Fun$($s$0)=$Fun$($s$)$ o $0$ for any s in $\{0,1\}*.$

Fun$($s$1)=1$ o Fun$($s$)$ for any s in $\{0,1\}*.$

Does Fun preserve regularity? i$.$e$.,$ is Fun $($L$)$ regular for every regular language L $?$

If yes, provide your reasoning, if not, provide an L for which Fun$($L$)$ is not regular.