Consider the function $f(z)=z+z^{*}$ where $z$ is a complex variable and $z^{*}$ denotes its complex conjugate. Which one of the following is TRUE?
1. $f(z)$ is both continuous and analytic
2. $f(z)$ is continuous but not analytic
3. $f(z)$ is not continuous but is analytic
4. $f(z)$ is neither continuous nor analytic