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Consider a function $f : \mathbb{R}\rightarrow\mathbb{R}.$ Say that $f$ is big if every point of the plane lies on the graph of $f$ or a tangent line to $f.$ What can we say about big functions? Is there already a name for this property?

I've got arguments that show that all odd degree polynomials of degree larger than $1$ are big, and that no other polynomials are big, but I've got no idea what you could say about general functions with this property.

  • Is there anything specific you are curious about? Without a specific question, this might be closed as being too broad of a question or that it is unclear what you're asking. – Clayton Oct 13 '18 at 16:38
  • @Clayton If I had to make it specific, I have a conjecture that this property is equivalent to $f''$ having a range of $\mathbb{R}$, but I'm curious if anything nice can be said. –  Oct 13 '18 at 16:41
  • I'd also be interested in knowing whether or not this property has a name--I've edited the question to include that. –  Oct 13 '18 at 17:00
  • Have you tried to prove that if $f$ is twice differentiable and $f'' > 0$ (or $f'' < 0$) then $f$ cannot be big? I am not sure this always holds, but would be a starting point fitting with my intuition. – Gibbs Oct 13 '18 at 17:14
  • I think your conjecture is not correct. Consider $f(x)=x + \sin x$. It's second derivative $f''(x)=-\sin x$ is bounded, but looking at the graph suggests that every point in the plane is covered by a tangent of $f$. – Ernie060 Oct 13 '18 at 17:51

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