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 Nov13 accepted A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity? Nov13 comment A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity? Actually, for my purpose, I need the space over all finite-dimensional vector spaces, but the generalization from the one-dimensional case is straightforward. Nov13 comment A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity? @PhoemueX Thanks for the positive answer. The original context for which I invented the space was that I wanted to construct a contravariant smooth functor in the sense of Serge Lang's §III.4 in his book Differential Manifolds. I wanted the functor be such that via Lang's construction one can associate an infinite-rank Banach vector bundle with the tangent bundle of a finite-dimensional smooth manifold. It is easy to construct such covariant functors as well as contravariant functors in the category of Fréchet spaces which lead to vector bundles with non-Banach Fréchet fibres. (cont.) Nov8 comment A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity? In view of Daniel Fischer's comment above, I added the "complex-analysis" tag. Nov8 revised A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity? deleted 1 character in body; edited tags Nov6 comment A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity? @Nishant An example is $t\mapsto t^{-1}\,\sin(t^2)$ . Nov6 asked A smooth nonzero function $\mathbb R\to\mathbb R$ with uniformly bounded derivatives tending to zero at infinity? Jul2 awarded Curious Mar20 accepted Are constants the only continuous functions with “symmetric derivative” zero? Mar20 comment Are constants the only continuous functions with “symmetric derivative” zero? @ Umberto P.: I already found it! The result I need is contained in Theorem 1.4 on page 6 and Corollary 1.5 on page 7. Thanks! Mar20 comment Are constants the only continuous functions with “symmetric derivative” zero? @ Umberto P.: From books.google.fi/…, I can see the table of contents. Could you please be a bit more specific? There is section 1.7 "Borel symmetric derivative". Is it possibly there? Mar20 comment Are constants the only continuous functions with “symmetric derivative” zero? In fact, in a way, I extracted it thereof! Mar20 comment Are constants the only continuous functions with “symmetric derivative” zero? @ geodude: Your deduction does not prove differentiability of $f$ . It is unfounded to write $2\lim_{z\to 0} \frac{f(r+z)-f(r)}{z} = 2\lim_{z\to 0} f'(r)$ since differentiability of $f$ is not known. Note that your $r$ depends on $s$ , hence on $z$ . Mar20 asked Are constants the only continuous functions with “symmetric derivative” zero? Dec8 comment Riemann integrability in not sequentially complete LCS? The question has been answered in mathoverflow.net/questions/151148/… . Nov17 asked Riemann integrability in not sequentially complete LCS? Oct28 awarded Tumbleweed Oct21 awarded Commentator Oct21 comment Series converging almost everywhere. See e.g. Richard M. Dudley's book Real Analysis and Probability (or any other similar) the pages where he proves completeness of the $L^p$ spaces (if I recall correctly) Oct21 answered Why do some people use $+\infty$ instead of $\infty$?