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The sum of a real power series is continuous at every point strictly inside its interval of convergence
Statement
If for , then is continuous at every satisfying .
Facts & Assumptions
Given: A power-series sum and a point strictly inside its radius.
The series converges uniformly on each closed interval strictly inside its radius (A power series converges absolutely and uniformly on every closed interval strictly inside its interval of convergence).
Every polynomial partial sum is continuous, since constants, the identity, powers, scalar multiples and finite sums are continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).
A uniform limit of continuous real-valued functions is continuous (The uniform limit of continuous real-valued functions on a metric space is continuous).
Proof
Choose so small that lies strictly inside .
The polynomial partial sums are continuous on this interval by [L2] and converge uniformly there to by [L1].
By [L3], is continuous on that interval, and in particular at .
Depends on
A power series converges absolutely and uniformly on every closed interval strictly inside its interval of convergence
The uniform limit of continuous real-valued functions on a metric space is continuous
Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function