A spline is a piecewise continuous, piecewise differentiable, etc. used to approximate more complicated functions. How differentiable depends upon what kind of spline you are using.
For example, a "linear spline" is just a piecewise linear function that is continuous at the "knots"- where the linear pieces connect. Obviously, you can't require it to be differentiable there without making it just a single straight line.
A "quadratic spline" consists of quadratic functions between knots that are both continuous and differentiable where they connect- but not twice differentiable.
A "cubic spline" consists of piecewise cubics that are twice differentiable where they connect.
Cubic splines are most commonly used. In fact, the name "spline" comes from the use of "splines"- very thin flexible strips of wood used to draw complex curves (before computer design). One can show by stress arguments that they are cubic splines.
The term b-spline comes from "basis" spline. If you think of the set of all possible splines (of a particular type:quadratic, cubic, etc.), you can show that that forms a vector-space and so any such spline can be written in terms of some basis.
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling We Value Civility
• Positive and compassionate attitudes
• Patience while debating We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving