I am trying to solve the Brachistochrone problem.
First of all I was asking myself whether there is any good reason that the curve $ x $ we are looking for is in $C^2[0,t_\text{end}] $. Then most textbook argue, that one could bring a curve of the form $x(t)=(x_1(t),x_2(t))$ to the form $x(t)=(x_1(t), f(x_1(t)))$ in order to obtain the integral that we want to examine. As far as I know, one would need that $x_1(t)$ is continuously differentiable and that $\forall t \in (0,t_\text{end}]: \dot{x_1}(t) >0 $ to do this but how do I know that there are no curves that are much faster that do not fulfill this property. for instance the curve where the ball falls free for several seconds is not included in this set, as this curve cannot be written as a function. but what is the right argument that one does not have to consider these curves? i have looked through so many textbooks but they are all fairly inaccurate about these things.
You know that $x_1'(t)\ge 0$. If for some $t$ you had $x_1'(t)=0$ when looking for a solution among functions $x_2=f(x_1)$ you would not find a solution. In fact any such curve (with vertical parts) could be approximated by graphs.