$$f(x)= \left\{\begin{array}{ll} x-x^2 &\mbox{if $x$ is rational,}\\ x+x^2 &\mbox{if $x$ is irrational.} \end{array}\right.$$ Show that $f'(0)=1$ and yet there is no neighborhood $I$ of the point $0$ on which this function is monotonically increasing.
$f(x)=x-x^2$ if $x$ is rational, $x+x^2$ if $x$ is irrational
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analysis