Is there a function satisfying as follows?

Question

I have a question. Is there a differentiable function $f:\mathbb{R}\to \mathbb{R}$ such that $c_1|x|\leq (f(x))^2\leq c_2 |x|$ for some positive constant $c_1,c_2$, and $$ \int_0^{\infty}(f'(x))^2dx<\infty$$ ?

Answer 1

In fact, isn't a differentiable function in $\Bbb R$ with $c_1|x|\leq (f(x))^2\leq c_2 |x|$ because this implies $f(0) = 0$ and $f'(0)$ does not exists: $$ \left|\frac{f(x) - f(0)}{x - 0}\right| = \left|\frac{f(x)}{x}\right|\ge\frac{\sqrt c_1}{\sqrt{|x|}}. $$