Interpolation, Extrapolation and Approximations rigorously A foreign book mentioned that 

"when the Lagrange's interpolation formula fails (for example with large sample due to Runge's phenomenon), you should use approximation methods such as Least-squares-method."

I am confused because I have always thought that interpolation/extrapolations are approximations. My confusion lies in the fact that the book used the three terms as disjoint while I have considered the first two terms as approximating. 
For example, can the Lagrange polynomial (also known as Lagrange interpolation) be extrapolation, interpolation and approximation at the same time? I would say yes and cannot see no problem to use the L polynomial to create extrapolations and approximations (I feel the terms fuzzy).
So how do you define the terms about approximation more rigorously?

Interpolation or extrapolation produces an exact formula, not an approximation, for a polynomial that matches given data.  However, this might be used as an approximation to the unknown function that produced those data.  
