A character on $C\left( \mathbb{R}^n\right)$ (the set of all complex-valued continuous functions on $\mathbb{R}^n$) is a continuous $^*$-algebra homomorphism into $\mathbb{C}$. For any fixed $x_0\in \mathbb{R}^n$, the function $\widetilde{x_0}:C\left( \mathbb{R}^n\right) \rightarrow \mathbb{C}$ defiend by $\widetilde{x_0}(f)=f(x_0)$ is a character.
Are there any characters of $C\left( \mathbb{R}^n\right)$ not of this form?
Thanks much in advance!
EDIT: Just to clarify, a $^*$-algebra homomorphism preserves addition, scalar multiplication, multiplication, the involution, and sends $1$ to $1$.
Since $\mathbb C[x_1,\ldots,x_n]$ is dense in $C(\mathbb R^n)$, it is sufficient to show that every character of $\mathbb C[x_1,\ldots,x_n]$ is of the form $\widetilde{x_0}$. But this is easy to see: if $\phi$ is any character, let $a_i = \phi(x_i)$. Then $\phi(f) = f(a_1,\ldots,a_n)$ since $\phi$ is a homomorphism.