A piecewise linear function is a continuous function that is an affine function on each cell of a polyhedral subdivision of its domain. For a function of one real variable, the cells are intervals and the graph is a polygonal curve. The breakpoints of the intervals need not be equally spaced.
For example, let 
.
Linear interpolation of the values 
gives a piecewise linear approximation to 
on ![[1,2]](/images/equations/PiecewiseLinearFunction/Inline4.svg)
.
The approximation improves as the mesh size of the partition
tends to zero. The trapezoidal rule integrates
this piecewise linear interpolant.
For a continuous piecewise linear function 
with breakpoints 
,
put 
and 
. The exact arc
length of its graph is
 |
For a smooth function, the lengths of its piecewise linear interpolants converge under suitable refinement to
 |
The graph of a piecewise linear function of two variables on a polygonal domain is a piecewise-linear surface. The one-variable graph is its one-dimensional analog.
