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Piotr Hajłasz authored notes on functional analysis developing Banach spaces, Hilbert spaces and the Baire category theorem. One application proves that a smooth real function is a polynomial if, at every point, at least one of its derivatives vanishes. The proof illustrates how a countable union of local conditions can force a global algebraic conclusion.^[ref]

If $f\in C^{\infty }(\mathbb{R})$ and every $x$ has some nonnegative integer $n=n(x)$ with $f^{(n)}(x)=0$, then $f$ is a polynomial. The derivative order may vary with $x$; there is no assumed uniform bound.^[ref]

Let $\Omega$ be the open set where $f$ is locally a polynomial. The closed sets $E_n=\{f^{(n)}=0\}$ cover $\mathbb{R}$. Applying the Baire category theorem on every closed interval shows that $\Omega$ is dense. Overlapping polynomials agree, so each connected component of $\Omega$ carries one polynomial.

If $X=\mathbb{R}\setminus \Omega$ were nonempty, it would be closed with no isolated points: adjacent polynomials at an isolated point have matching derivatives and glue across it. Apply the Baire category theorem on $X$ to find an open interval $I$ with $\varnothing \ne X\cap I\subset E_n$. Taking derivatives along accumulation points gives $f^{(m)}=0$ on $X\cap I$ for all $m\ge n$. Every intervening polynomial component has an endpoint there; matching its derivatives forces its degree below $n$. Hence $f^{(n)}=0$ throughout $I$, contradicting $X\cap I\ne \varnothing$. Thus $\Omega =\mathbb{R}$, and overlapping local polynomials give one global polynomial.