Real-valued function

A real-valued function has codomain contained in the real numbers.

A global maximum of a real-valued function is a value $f(x_0)$ satisfying $f(x)\le f(x_0)$ throughout its domain.

A global maximum is one type of extremum of a real-valued function; global minima reverse the inequality.

The positive part of a real-valued function $f$ is $f_{+}(x)=\max (0,f(x))$. Together with its negative part, it gives $f=f_{+}-f_{-}$ and $|f|=f_{+}+f_{-}$.

The negative part of a real-valued function $f$ is the nonnegative function $f_{-}(x)=\max (0,-f(x))$. It records the magnitude of the negative values, rather than retaining their sign: if $f(x)=-3$, then $f_{-}(x)=3$. Together with the positive part, it gives $f=f_{+}-f_{-}$ and $|f|=f_{+}+f_{-}$. This decomposition defines the Lebesgue integral of a signed function and makes the integrability condition explicit.