# Linearity interval

When a univariate function is piece-wise linear, it has the form $LaTeX: a_i x + b_i$ for $LaTeX: x$ in the interval $LaTeX: [c_i, c_{i+1}]$, where $LaTeX: a_i$ is not equal to $LaTeX: a_{i+1}$. (The phrase usually means the function is continuous.) This arises in linear programming when considering the optimal value as a function of varying right-hand sides (or cost coefficients) in fixed proportions: $LaTeX: b+td$ (or $LaTeX: c+td$), where $LaTeX: d$ is an admissible change vector and $LaTeX: t$ is the (scalar) variable. Then, for the range of $LaTeX: t$ where the LP has a solution, the optimal value function has this piece-wise linear (continuous) form, and the intervals that comprise the domain are called linearity intervals.