# Breakpoint

A point where some property of a function changes, such as its slope. For example, a piece-wise linear, univariate function with $LaTeX: k$ breakpoints has the form:
$LaTeX: f(x) = a_i x + b_i \mbox{ for } c_i \le x < c_{i+1}$
where $LaTeX: i=1, ...,k$ and $LaTeX: c_{k+1}$ is defined to be infinity and $LaTeX: c_1$ is the least value of $LaTeX: x$. This function is continuous if it is equal at breakpoints, but the slope changes if $LaTeX: a_i \neq a_{i+1}$ for $LaTeX: i=1, ..., k-1$.