TL:DR, if you have a function centered at "a" with the derivative at "a" being positive, can the function not be increasing around a? What does that mean about points around "a"?

I had a problem given to me in a math tutorial recently that asked:

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> Construct a function f satisfying all the following properties:
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> •Domain f = $\mathbb{R}$
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> •f is continuous
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> •$f^{\prime}(0)=0$
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> •f does not have a local extremum at $0$.
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> 
> •There isn’t an interval centered at $0$ on which $f$ is increasing.
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> 
> •There isn’t an interval centered at $0$ on which $f$ is decreasing.
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> 
> 

The function we were shown that worked was $f(x) = \displaystyle \lim\_{c \to x}c^{2}\sin(\frac{1}{c})$.

I was wondering if instead, the derivative of some number "$a$" was strictly larger than zero ($f^{\prime}(a)>0$) and the extremum was at "$a$" and the intervals centred at "$a$" if a function could be created to meet the criteria. Can you think of criteria that could be removed to make this work if it doesn't? Would love to learn more about these types of derivative questions, since this one was quite confusing and interesting.