2022-16 Identity for continuous functions

For a positive integer \( n \), find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) such that
\[
\sum_{k=0}^n \binom{n}{k} f(x^{2^k}) = 0
\]
for all \( x \in \mathbb{R} \).

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