Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 4 down vote favorite

Give a differentiable function that has a positive derivative at $0$, yet is not increasing on any open neighbourhood of $0$.

I believe that the required function needs to have a derived function that is discontinuous at $0$ (ie. the required function needs to be not continuously differentiable at $0$). Any suggestion would be appreciated.

share|improve this question
Nice title! $\hskip4in$ – JavaMan Feb 12 at 1:38
Functions of the form $Ax^\alpha+Bx^\beta\sin(x^\gamma)$ are your friends for these types of problems. – David Mitra Feb 12 at 1:43

2 Answers

up vote 2 down vote accepted

This is Example 3.5 in Gelbaum and Olmsteads' Counterexamples in Analysis:

The function $f$ defined by $$f(x)=\cases{x+2x^2\sin(1/x),& $x\ne0$\cr 0, &$x=0$}$$ is differentiable, $f'(0)=1$, but $f'$ takes both positive and negative values in every neighborhood of $0$.

share|improve this answer
Thanks! $f'(x)=1+4x\sin\frac1x-2\cos\frac1x$ in every punctured neighbourhood of $0$. How to show that this derived function takes both positive and negative values in every punctured neighbourhood of $0$ ? – Ryan Feb 12 at 2:53
@Ryan $f'(1/(k\pi) ) =1+0-2=-1$ if $k$ is even and $f'(1/(k\pi) ) =1+0+2=3$ if $k$ is odd. – David Mitra Feb 12 at 2:56
Brilliant! I don't know how I would've solved this problem on my own. Thank you. – Ryan Feb 12 at 3:01
@Ryan You're welcome. Glad to help. (I think you'd have eventually found it, using Ross' hint.) – David Mitra Feb 12 at 3:16
Insertion for super-explicitness: For each neighbourhood $N$ of $0$, there is some odd $k_1$ and some even $k_2$ such that $\frac 1{k_1\pi},\frac1{k_2\pi} \in N$. – Ryan Feb 12 at 3:30
up vote 2 down vote

Hint: you need something that wiggles faster and faster as it gets near zero, so it has a decreasing part in every neighborhood of zero. Can you think of a common example?

share|improve this answer
@Ryan Think about wiggling along the line $y=x$. – David Mitra Feb 12 at 1:55

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.