Let $f$ be a non-constant, complex-valued function which is defined and analytic for all $z$ in the complex plane. Also, $f$ has the additional property that it is always real. To me, such a function seems bizarre and unlikely. Does such a function exist?
Using the Cauchy–Riemann equations, as mentioned in the comments, is probably the most elementary way. Here are two other possibilities.
If you can use Liouville's theorem, then $g(z)=\exp(i\,f(z))$ is an entire bounded function and hence constant. This means that $f$ takes values in a discrete set. Since $f$ is continuous, it must be constant.
If you can use the open mapping theorem, then the image of $f$ is not open and so $f$ must be constant.