A piecewise-linear function is linear on each member of a finite or locally finite partition of its domain into intervals or polyhedral pieces.
Between two prescribed values, linear interpolation assigns at , for . Joining successive samples produces a piecewise linear function. For a continuous function on a compact interval, interpolants along grids with mesh tending to zero converge in the uniform norm; their error is bounded by the modulus of continuity at the mesh size.
A linear interpolant is the unique polynomial of polynomial degree at most one taking prescribed values at distinct interpolation nodes . It is an affine function, including the constant function case. For , its two displayed coefficients are nonnegative and sum to one, so changing each sampled value by at most changes every interpolated value by at most . Differentiation instead uses the difference of the two samples divided by , causing noise amplification by differentiation of a linear interpolant.
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A piecewise linear function is a function composed of multiple linear segments. Each segment is defined by a linear equation over a specific interval in its domain. Essentially, the function "pieces together" several lines to create a graph that can take various forms depending on the specified intervals and the slopes of the lines.