A semialgebraic function 
, where 
is a semialgebraic
set, is a function whose function
graph
 |
is a semialgebraic set (Plaumann 2023, p. 12). Polynomial functions are semialgebraic, as are rational functions on semialgebraic domains where their denominators do not vanish.
For example, the absolute value function 
is semialgebraic because its function
graph is described by 
and 
. A semialgebraic function need not be continuous.
The function equal to 0 for 
and to 1 for 
is a discontinuous example.
Continuous semialgebraic functions supply the finite bounds used in cylindrical algebraic decomposition.
