If a function \( f \) is analytic on the disc \( D \) and has at least one zero coefficient in its power series expansion at every point \( a \in D \), it can be concluded that \( f \) is a polynomial. The discussion emphasizes the use of the Heine-Borel Theorem to establish that within any closed subdisk \( D' \), the sets \( A_n \) of points where the \( n \)-th derivative of \( f \) equals zero must contain an accumulation point. This leads to the conclusion that if one of these sets, say \( A_m \), is infinite, then \( f^{(m)} \) must equal zero in \( D' \). The identity theorem supports the argument that if \( f^{(m)} \equiv 0 \), then \( f \) is indeed a polynomial. The discussion confirms the validity of these claims, reinforcing the connection between the properties of derivatives and the polynomial nature of \( f \).