What is the continuity of the composite function f(g(x)) given

What is the continuity of the composite function $f\left(g\left(x\right)\right)$ given $f\left(x\right)=\frac{1}{\sqrt{x}}$ and $g\left(x\right)=x-1$
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Explanation:
The process here is to verify first the domain of g(x) (the inner function in the composition).
The domain is $\left\{x\mid x>1,x\in \mathbb{R}\right\}$
$f\left(g\left(x\right)\right)=\frac{1}{\sqrt{x}}$
There will be two types of restriction on the domain in this problem.
- When the number underneath the $\sqrt{}$ is less than 0.
- When the denominator equals 0.
The number under the square root will be negative whenever $x<1$. The denominator will equal 0 when $x=1$, so the domain of the composition is
$\left\{x\mid x>1,x\in \mathbb{R}\right\}$