Say there is a continuous function $f(x)$ which does not have any undefined points in the interval $[a,b[$. I want to randomly select a point $(x,y)$ in that interval on the line formed by that function.
This seems to be no easy task, as simply picking a random value of $x$ and thus getting a point will not give a "fair" chance in every possible case to all of the points in the interval on the actual function. Naturally, this is the case because the derivative can vary in different points in that interval. From that too, the "length" of the line of the function in that interval will also be longer than the interval itself.
The closest thing I can imagine to this would be to literally travel on an imaginary boat across the range of the line defined by the interval up to a randomly selected point on its length. To that extent, we could calculate the arc length of that interval, pick a random point in that range, and then convert that point to a point on the curve. However, the last process just brings us a whole new problem.
Just for reference, yes $f(x), [a,b[$ has to be generalized. Any function (which is not piecewise) satisfying the aforementioned conditions is being considered here.
Thank you for your help!