Entire function with vanishing derivatives? Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function.
And assume that at each point, one of it's derivatives vanishes.
What can you say about $f$?
A hint suggests that $f$ must be a polynomial.

we can also solve this without using baire category theorem..
we can  use the fact that zeros of non zero analytic  function is countable. Note that union of $A_{n}$ is  $\mathbb{C}$ thus we must have some fix derivative identically 0. Now use power series and analytic continuation.
