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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?

- $f(z)$ is both continuous and analytic
- $f(z)$ is continuous but not analytic
- $f(z)$ is not continuous but is analytic
- $f(z)$ is neither continuous nor analytic

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