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 \frac1e\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
`\frac1e<x<e`.