Is the function $y=x/2 + x^{2}\sin(1/x)$ monotonic near $0$?
The derivative $f'$ obviously goes positive and negative near $0$, because $f'(x)= \frac12 + 2x\sin(1/x) - \cos(1/x))$ Does that mean that $f$ is not monotonic near $0$?
Is the function $y=x/2 + x^{2}\sin(1/x)$ monotonic near $0$?
The derivative $f'$ obviously goes positive and negative near $0$, because $f'(x)= \frac12 + 2x\sin(1/x) - \cos(1/x))$ Does that mean that $f$ is not monotonic near $0$?
It may be appropriate to now summarize what is spread across comments into an answer.
The function $f(x) = \begin{cases}\tfrac x2+x^2\sin(1/x)&\text{if }x\ne 0\\ 0&\text{if }x=0\end{cases}$ has one property that may make it look like being monotonic near $0$: For $0<|x|<\frac 12$ we have $|x^2\sin(1/x)|\le|x^2|<\left|\frac x2\right|$, hence $f(x)>0$ for $0