There is a function: $f: \mathbb{R} \to [1 ; + \infty) $ that is continous in $\mathbb{R}$. Prove that for every chosen set of $x_i \in \mathbb{R}$ for $i \in \{1, 2, ..., n \}$ there exists a point $x \in \mathbb{R}$ such that:
$$f(x) = \sqrt[n]{f(x_1) \cdot f(x_2) \cdot f(x_3) \cdot ... \cdot f(x_n)}$$
That exercise seams quite basic: if function is continuous in $\mathbb{R}$, then, for every set of values, there exists a value equal to geometric mean of values in the set that is reached by that function.
I know that geometric mean is a continous function + that a function $f(x)$ is continous. How do I prove then that a combination of those functions is continous?
As I carry on my research, any help would be much appreciated - answers providing materials on similar problems would be great.