Monotonicity in an interval

Let $f:[a,b]\rightarrow R$ be differentiable at $c\in (a,b)$ with $f'(c)<0$. Does this imply that $\exists$$\delta>0 such that f is monotonic in (c-\delta,c+\delta)? - 4 Answers Define f to be x+2x^2\sin\frac{1}{x} for x\neq 0 and 0 at x=0. This is an example of a function that has a positive derivative, but is not monotonic on any open interval surrounding that point. Now just multiply it by -1. - The short answer is: no. Here is an example:$$ f(x) = -x^2(2+\sin(1/x)), \qquad f(0)=0 $$- Define f:(-1,1)\rightarrow\mathbb{R} by$$ f(x) = \left\{ \begin{array}{rl} -\sin(x) &\mbox{ if$x$is rational} \\ - x &\mbox{ otherwise} \end{array} \right.$\$

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There are even differentiable functions that are not monotone on any interval, as seen in a previous question.

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