Question

How do you determine where f(x)=cos^(-1)(lnx) is continuous?

Answer 1

$\ln x$ is continuous over its domain, all real $x>0$.

Meanwhile, $\cos^{-1}y$ is defined for real $-1\le y\le1$.

If $y=\ln x$, then we have $-1\le \ln x\le1\implies \dfrac1e\le x\le e$ as the domain of $\cos^{-1}(\ln x)$.

We know that if $f$ and $g$ are continuous functions, then so is the composite function $f\circ g$.

Both $\cos^{-1}y$ and $\ln x$ are continuous on their domains (excluding the endpoints in the case of $\cos^{-1}y$), which means $\cos^{-1}(\ln x)$ is continuous over $\dfrac1e$.