Continuity of an exponential series

Question

Let $(a_n)_{n \in \mathbb{N}}$ be an infinite positive sequence of integers. Assume that $f(\lambda) = \sum\limits_{n=0}^{\infty} a_n \lambda^n$ is finite for any $\lambda \in [0, \lambda_0)$. Does this necessarily imply that $f(\lambda)$ is also continuous in $(0, \lambda_0)$, or might something strange happen?

Answer 1

If your series converges for $\lambda=r$, then it converges uniformly for all $\lambda\in(-r,r)$. This is because for such $\lambda$ $$ \sum_{n>m}a_n|\lambda|^n\leq\sum_{n>m}a_nr^n\xrightarrow[m\to\infty]{}0. $$

So $f(\lambda)$ is a uniform limit of polynomials. And a uniform limit of continuous functions is continuous.

Your function is in fact a lot more than continuous. It is analytic.