Terminology: Spline interpolation I have read two different definitions of Splines:

  
*
  
*A differentiable piecewise polynomial.
  
*A piecewise polynomial.
  

If I build a piecewise polynomial using cubic polynomials, it's differentiable, and thus we could call it a "Cubic Spline". The same happens if I use quadratic polynomials.
However, when it comes to using linear polynomials the two definitions diverge. Under the first one it wouldn't be a spline, since it doesn't have enough conditions to be differentiable. Under the second one, we could call it a "Linear Spline".
Which of these two definitions is right ? Does it make sense to say "Linear Spline" ?
 A: Different people use different terminology.
One of the leading experts in the area is Carl de Boor. He uses the term "spline" to refer to any piecewise polynomial, regardless of its differentiability. So, for de Boor, piecewise linear functions and even piecewise-constant step functions are still "splines". For more details, refer to his book "A Practical Guide to Splines".
Other people use the term spline to refer to a piecewise polynomial of degree $m$ that has a continuous derivative of order $m-1$. The folks who wrote this Wikipedia page, used this definition. So, for these people, a cubic spline (degree 3) is a piecewise cubic function that is $C_2$. Of course, piecewise polynomials with fewer continuous derivatives are often useful, too. For example $C_1$ piecewise cubics (e.g. Catmull-Rom splines) are very useful. These "less smooth" functions are sometimes known as "subsplines".
Between the two usages, I think the first (de Boor's) is more common. So, for most people, a "spline" is simply a piecewise polynomial.
It's also worth mentioning that polynomial splines are not the only kind. Splines are often constructed from rational functions, and sometimes even from trigonometric or exponential functions.
