This question was asked in a masters exam for which I am preparing and I was unable to think about it. So, I am asking it here.
Question: Let f be a continuous function from $\mathbb{C} \to \mathbb{C} $ such that $f(z^2+2z-6) $ is an entire function. Show that f is an entire function.
As fog is entire where g is $ z^2+2z-6$ . So,$ (f'og )\times g' $ exists for all but how can I use it to verify analyticity of f?
Please help.
Thanks!!
Hint: Let $g(z)=z^2+2z-6$. Now, $g'(z)\not=0$ for all $z\not=-1$. So, for each $z\not=-1$ we can find two small open sets $U_z$ and $V_z$ such that $g\big|U_z\to V_z$ is bi-holomorphic. So, considering $\big(f\circ g\big)\circ \big(g\big|U_z\to V_z\big)^{-1}$ we can say $f$ is holomorphic on $\Bbb C\backslash g^{-1}(-1)$. Now, $f$ is continuous on $\Bbb C$, so by Riemann removable singularity theorem $f$ is holomorphic. Note that $g^{-1}(-1)$ can have at most two elements.