real-analytic function
Some functions are so well-behaved that, near any point, they ARE their own power series — not merely approximated by it, but exactly equal to it on a small interval. These are the real-analytic functions, the gold standard of smoothness on the real line. Polynomials, the exponential, sine, cosine and their combinations all belong to this elite club.
Precisely, a function f defined on an open set is real-analytic if for every point a in the set there is a radius r > 0 and coefficients c_n such that f(x) = sum_{n=0}^infinity c_n (x - a)^n for all x with |x - a| < r. The coefficients are forced to be the Taylor coefficients f^{(n)}(a)/n!, so being analytic means the Taylor series at every point actually converges back to the function.
Real-analyticity is strictly stronger than being smooth (infinitely differentiable). Every analytic function is smooth, but the converse fails: there are smooth functions that are not analytic at a point because their Taylor series, though defined, does not represent them. Analytic functions are also rigid — knowing one on a tiny interval determines it everywhere it can be continued, a phenomenon explored under analytic continuation.
A power series, within its radius of convergence, is automatically analytic — not only at the center but at every interior point, and re-expanding around a new center gives a possibly larger disk. This is why analytic functions chain together so smoothly.