Let $f(x) = \sum_{n=0}^{\infty} a_n x^n$; suppose $f$ has a radius of convergence $R\in [0,\infty]$. Power series are uniformly continuous on the interior of their disk of convergence, permitting among other operations term-by-term differentiation and integration. However, even if ${R}=\infty$, I have only ever seen results that allow term-by-term operations on bounded intervals. This is usually sufficient but I wanted to double-check why this is the case.
My intuition is that the only polynomials that are uniformly continuous on all of $\mathbb{R}$ are linear or constant polynomials: this can easily be proved from the definition of uniform continuity. So even if the series converges everywhere, if it is 'non-linear' in the sense that $a_k\ne 0$ for some $k\ge 2$, then the partial sums are not uniformly continuous on $\mathbb{R}$, which means the operations must be done 'locally' on a bounded interval. Is this correct, roughly speaking?