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First, I would like to find a function $f:\left[ 0,1 \right] \rightarrow \mathbb{R}$ such that $f$ is continuous but not monotonic in any interval. Secondly, I want to find a, continuous yet not monotonic in any interval, function $g:\mathbb{R} \rightarrow \mathbb{R} $.

Since a function is not continuous in any interval $\leftrightarrow$ the function is not continuous in any interval with rational edges, I thought about finding a function similar to dirichlet function, but couldn't find one that would suit me. Any suggestions? Or maybe I'm looking for the wrong "type" of functions?

Roman Vale
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  • Is the only difference $f$ and $g$ that $f$ has domain $[0,1]$? – Caleb Stanford Apr 20 '14 at 16:17
  • See the answers here. http://math.stackexchange.com/questions/719644/are-continuous-functions-monotonic-for-very-small-ranges/719674#719674 – Seth Apr 20 '14 at 16:18
  • http://en.wikipedia.org/wiki/Weierstrass_function and http://en.wikipedia.org/wiki/Blancmange_curve – Seth Apr 20 '14 at 16:20

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If a function is monotonic on an interval it is differentiable almost everywhere on that interval, so it suffices to find a continuous nowhere differentiable function. One such example is the Takagi function. This gives you the function $f$ you are looking for. For the function $g$ just paste together a bunch of copies of $f$.

Seth
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