Property of a continuous function in the neighborhood of a point

If you take $f(x)=\begin{cases}x^2\sin(1/x),& x\neq0\\0,& x=0\end{cases}$ and take $a=0$, then for all $x>0$, there are values of $t_1,t_2,t_3\in(0,x]$ such that $f(t_1)>0=f(0)$, such that $f(t_2)=0=f(0)$ and such that $f(t_3)<0=f(0)$, respectively.

Therefore $M,N,P$ are empty.

Observe that $f$ is even differentiable at all points. Therefore, even for differentiable it is not true.

One can even have $f$ smooth. For example, replace the $x^2$ above with $e^{-1/x^2}$ and the function will be smooth, but still have the same oscillatory behavior near $x=0$.

This is not true in general! Consider

$$ \begin{aligned} f(x):= \begin{cases} x\sin(1/x)&\text{ if }x\not =0\\ 0 &\text{ if }x=0 \end{cases} \end{aligned}. $$

This function is continuous at the point $0$, but it oscillates indefinitely as it approaches $0$. Hence, all three of your sets will be empty.