If any function \kappa is given, is there always exist a function which its curvature function is \kappa?

Question

Let $\kappa 2 : (a,b) \subset \mathbb{R} \rightarrow \mathbb{R} $ be a arbitrary function.

Then is there always exist a function satisfies

$ {f''(x) \over (1+(f'(x))^2)^{3/2}} = \kappa 2(x) $ ?

I thought about a function

$ \kappa 2(x) = \begin{cases} & 1 \text{ if } x \in \mathbb{Q} (\subset (a,b)) \\ & 0 \text{ if } x \notin \mathbb{Q} (\subset (a,b)) \end{cases}$

But I couldn't find the way to prove that there is no function satisfies above conditon.

Answer 1

As $f''$ exists, $f'$ is continuous. $f''$ is a derivative and derivatives have the Darboux property. Prove that the LHS has the Darboux property and you have a contradiction.