All derivatives zero at a point $\implies$ constant function? Suppose $f: \mathbb{R} \rightarrow \mathbb{R} $ is a continuous function, and there exists some $a \in \mathbb{R}$ where all derivatives of $f$ exist and are identically $0$, i.e. $f'(a) = 0, f''(a) = 0, \ldots$ Must $f$ be a constant function? or if not, are there examples of non-constant $f$ that satisfy these properties? 
What if the hypothesis is changed so that the derivatives of $f$ are identically $0$ on an open interval, i.e. $f'(A) = 0, f''(A) = 0, \ldots$ for some open interval $A$. Are the answers still the same?

Yes, such functions do exist; they’re called flat functions. The simplest example that I know is the one given at the link:
$$f(x)=\begin{cases}
e^{-1/x^2},&\text{if }x\ne 0\\
0,&\text{if }x=0\;,
\end{cases}$$
which is flat at $x=0$.
You can modify this example to get one that is flat on the interval $[0,1]$ but not constant on $\Bbb R$:
$$f(x)=\begin{cases}
e^{-1/x^2},&\text{if }x<0\\
0,&\text{if }0\le x\le 1\\
e^{-1/(x-1)^2},&\text{if }x>1\;.
\end{cases}$$
In effect I’ve just cut the function at $x=0$ and moved the righthand half $1$ unit to the right, filling in the gap with the zero function.
