Interpolation in general is a way to determine intermediate values from a set of coordinates. Linear interpolation would be to fit a single linear function to the entire set of coordinates. Piecewise linear interpolation would then be to determine intermediate values from the set of coordinates by fitting linear functions between each set of coordinates. Connect-the-dots so to speak.
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Linear interpolation is used as a method used in mathematics of constructing a curve that has the best fit to a series of points of data using linear polynomials.
The form of the piecewise functions can be arbitrarily complex, but higher degrees of specification require considerably more user input.
Piecewise, linear, exponential, quadratic, Onto, cubic, polynomial and absolute value.
What you are asking is not precisely clear, but in general missing data is filled in by a process of interpolation. eg. Linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
f is a piecewise smooth funtion on [a,b] if f and f ' are piecewise continuous on [a,b]