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The aim of the paper is to introduce and study a new class of spaces called mildly ${(1,2)}^{*}$-normal spaces and a new class of functions called ${(1,2)}^{*}$-$\mathrm{rg}$-continuous, ${(1,2)}^{*}$-$\mathrm{R}$-map, almost ${(1,2)}^{*}$-continuous function and almost ${(1,2)}^{*}$-$\mathrm{rg}$-closed function in bitopological spaces. Subsequently, the relationships between mildly ${(1,2)}^{*}$-normal spaces and the new bitopological functions are investigated. Moreover, we obtain characterizations of mildly ${(1,2)}^{*}$-normal spaces, properties of the new bitopological functions and preservation theorems for...