Evaluation character (${\delta }_x$)

For an algebra $A$ of complex-valued functions on a set, evaluation at $x$ is the multiplicative linear functional ${\delta }_x(f)=f(x)$. It is an algebra character precisely when it is nonzero, meaning that some $f\in A$ has $f(x)\ne 0$. This holds automatically if $A$ contains the constant function one.

The nonzero requirement matters for algebras with a common zero: evaluating $\{f\in C[0,1]:f(0)=0\}$ at zero gives the zero linear functional, which is not an algebra character.