[tex]\ln x[/tex] is continuous over its domain, all real [tex]x>0[/tex].
Meanwhile, [tex]\cos^{-1}y[/tex] is defined for real [tex]-1\le y\le1[/tex].
If [tex]y=\ln x[/tex], then we have [tex]-1\le \ln x\le1\implies \dfrac1e\le x\le e[/tex] as the domain of [tex]\cos^{-1}(\ln x)[/tex].
We know that if [tex]f[/tex] and [tex]g[/tex] are continuous functions, then so is the composite function [tex]f\circ g[/tex].
Both [tex]\cos^{-1}y[/tex] and [tex]\ln x[/tex] are continuous on their domains (excluding the endpoints in the case of [tex]\cos^{-1}y[/tex]), which means [tex]\cos^{-1}(\ln x)[/tex] is continuous over [tex]\dfrac1e<x<e[/tex].
