Prove convexity of function
$$f(x_1, x_2 \dots x_n):=\dfrac{1}{x_1 - \dfrac{1}{x_2 - \dfrac{1}{\ddots - \dfrac{1}{x_n}}}}$$
defined on subset of $\mathbb{R}^n$, where every denominator is strictly positive.
I was thinking about induction (and I would like not to try to differentiate that). Also I was trying to use proven fact from this link:
Midpoint-Convex and Continuous Implies Convex
But it works only when $f$ is continuous, which we seem not to have here.
