Real analytic function

A function $f$ is real analytic at $y$ when its Taylor seriesconverges to $f(x)$ throughout some neighborhood of $y$.

A smooth function on a real open interval is real analytic exactly when, near each point, there are constants $C,A>0$ such that $|f^{(n)}(x)|\le C{A}^{n}n!$ for every $n$ and every $x$ in that neighborhood. The bound must hold on a neighborhood, not only at its center. The Taylor theorem with Lagrange remainder proves sufficiency; differentiating a convergent power series on a smaller interval proves necessity. A flat function shows why pointwise bounds alone are insufficient.

Suppose a family of polynomial coordinate maps has a fixed bound on its degrees and coefficients depending continuously on a compact parameter set, and sends a fixed real compact set into a compact subset of the domain of a real analytic function. A common complex neighborhood can then be chosen so the compositions are holomorphic and uniformly bounded there. The multivariable Cauchy estimate gives uniform factorial derivative bounds. For the change $\tilde{x}=x+y^2-\epsilon$, the coefficient pullback uses its inverse: $f_{\epsilon }(\tilde{x},\tilde{y})=f(\tilde{x}-{\tilde{y}}^2+\epsilon ,\tilde{y})$.