Intereting Posts

Is $\left(\sum\limits_{k=1}^\infty \frac{x_k}{j+k}\right)_{j\geq 1}\in\ell_2$ true if $(x_k)_{k\geq 1}\in\ell_2$
Is there a better counter-example? (problem involving limit of composition of functions)
Does the open mapping theorem imply the Baire category theorem?
Which Cross Product for the Desired Orientation of a Sphere ?
Inequality: $(a^3+a+1)(b^3+b+1)(c^3+c+1) \leq 27$
Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational?
dense in terms of order and in terms of the order topology
How can we evaluate this $\prod_{k=1}^n(1+kx)$
Recurrence relation for the number of ternary strings containing 2 consecutive zeros vs not containing
Find the value of : $\lim\limits_{n\to \infty} \sqrt {\frac{(3n)!}{n!(2n+1)!}} $
Why $\dim U+\dim U^\perp=\dim V$?
Convert this equation into the standard form of an ellipse
What's the precise meaning of imaginary number?
The origin is not in the convex hull $\Rightarrow$ the set lies in a hemisphere?
A definite integral with trigonometric functions: $\int_{0}^{\pi/2} x^{2} \sqrt{\tan x} \sin(2x) \, \mathrm{d}x$

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function such that for every $x \in \mathbb{R}$ there exists $n$ such that $f^{(n)}(x) = 0$, then f is a polynomial.

I’m kind of lost on this one. I know that I have to use Baire’s category theorem somewhere here, but I’m not sure exactly how.

- Baire: Show that $f\colon \mathbb{R}\to\mathbb{R}$ is a polynomial in an open bounded set
- Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?
- Normal numbers are meager
- Infinite intersection between a arbitrary set of integers and a set of floor powers
- Is $$ a countable disjoint union of closed sets?
- Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

- Clopen subsets of $A^\Bbb N$ for finite $A$
- Generic Elements of a Set.
- Products of Continuous functions: Topology
- Homeomorphism of the real line-Topology
- The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.
- Uncountable closed set of irrational numbers
- A topological vector space with countable local base is metrizable
- Which of the following subsets of $\mathbb{R}^2$ are compact
- When is the ring of continuous functions Noetherian?
- Existence of a continuous function which does not achieve a maximum.

This problem can indeed be solved using the Baire Category Theorem.

A nice discussion of variants of the problem, along with an outline of a proof of your version, can be found in the answers to this post at MathOverflow.

- Quadratic variation – Semimartingale
- Should the domain of a function be inferred?
- Books for a beginner
- How to solve this trig integral?
- Find a matrix transformation mapping $\{(1,1,1),(0,1,0),(1,0,2)\}$ to $\{(1,1,1),(0,1,0),(1,0,1)\}$
- Least norm in convex set in Banach space
- Why do we not have to prove definitions?
- Representing the multiplication of two numbers on the real line
- Finite Group with $n$-automorphism map
- A geometric reason why the square of the focal length of a hyperbola is equal to the sum of the squares of the axes?
- Is the trace of inverse matrix convex?
- What would be the shortest path between 2 points when there are objects obstructing the straight path?
- $g(x) = 1/(1+x^2)$ is continuous everywhere epsilon delta approach
- Is there a formula for solving the congruence equation $ax^2 + bx + c=0$?
- Tensor products over field do not commute with inverse limits?