Do we always need to be given points to do interpolation? Or can we be given only a function? For lagrangian interpolation we require points, and does it apply for others also?
There are two points of view.
The first is when you have empirical data and you want to find the function by having only points. Then you use interpolation.
The second one is when you're given a function and you want to approximate its derivative or integral, for example. Then you choose points of the function yourself to apply the numerical methods. If all you have is empirical data and you want to approximate the derivative or integral, you apply the numerical methods obtained when you supposed you knew the function but this time you can't choose the points, they are imposed.