Is it true that if $f$ is injective, then it must be analytic?

Question

I'd be interested to know about the connection between injectivity and analyticity of functions. In my humble guess, if $f$ is a real or complex valued function such that it is injective, then it would satisfy the Cauchy–Riemann equations, which implies it is analytic. But this is a weaker reason, because I don't know if it's true that "if $f$ is injective then all its derivative exist". Hence, my question. Is it true that if $f$ is injective, then it must be analytic?

Answer 1

$f(z)=\overline {z}$ is a counterexample.