Fixed-point for generic continuous function $f:\mathbb{R} \to \mathbb{R}$ such that $\forall x, y \in \mathbb{R} \space|f(x) - f(y)| < \sqrt{|x - y|}$

Question

I'm trying to solve this real analysis problem. Is it true that, given a generic continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $\forall x, y \in \mathbb{R} \space |f(x) - f(y)| < \sqrt{|x - y|} $, it has a fixed point? The answer, that I know a priori, is yes, but I'm having difficulties proving that.

I thought about using the fixed-point theorem (Banach-Caccioppoli) but unfortunatly $f$ is not necessarily a contraction since if $f$ is also differentiable $(\forall x, y \in \mathbb{R}) ((x \ne y) \implies (\frac{|f(x) - f(y)|}{|x-y|}<\frac{1}{\sqrt{|x-y|}})) $ and, having fixed y, if $x \to y$, $f'(y) \le +\infty$, and nothing seems to assure that the derivative is bounded ($\exists c \in (0, 1) \space \forall x \in \mathbb{R} \space f'(x) < c$ should also hold).

Thanks in advance for any help

Answer 1

If $f$ has no fixed point, then $f(x)>x$ for all $x$ WLOG. Then $x<f(x)\le f(0)+{\sqrt x}$ for all $x>0$, absurd.