I have some issue on solving the following question.
Let D be an open disc of radius ε > 0 centered at the origin. Does there exist an analytic function $f : D → C$ , which takes the following values on a positive real axis: $f(x)=x^4sin(\frac{1}{x})$ for all $x∈R∩D, x>0$
I believe the answer is no. My attempt to prove: let $g(z)=z^4sin(\frac{1}{z})$ where g: D \ {$0$} -> C. I know that every point in the open disc D is a limit point. Let E be a set in D that contains a limit point. So we can apply the Uniqueness Principle which states that g(z) = f(z) for every z ∈ E. I am now stuck on proving the function f(z) = g(z).
The answer is indeed no, because otherwise we would have$$\bigl(\forall z\in D_\varepsilon(0)\setminus\{0\}\bigr):f(z)=z^4\sin\left(\frac1z\right).$$But there is no such analytic function, because $f\left(\frac1{\pi n}\right)=0$ if $n$ is large enough. So, $0$ would be a non-isolated zero of $f$, and non-constant analytic functions with a connected domain have no non-isolated zeros.