Continuity of Derivative at a point. Is it possible that derivative of a function exists at a point but derivative does not exist in neighbourhood of that point.
If this happens then how is it possible.
I feel that if derivative exists at a point then the left hand derivative is equal to the right hand derivative so derivative should exist in neighbourhood of that point.
 A: Yes, consider $f(x) =
\begin{cases}
x^2  & x  \in \mathbb{Q} \\
0 & x \in \mathbb{R} \setminus \mathbb{Q}
\end{cases}$
$f$ is differentiable at $0$ and nowhere else. 
A: This is not exactly an answer to your question,
but I think the source of your confusion
is that you seem to believe that the left/right hand derivatives
$$f'_\pm(a)=\lim_{h\to 0^\pm} \frac{f(a+h)-f(a)}{h}$$
are the same things as the left/right hand limits of the derivative
$$\lim_{h\to 0^\pm} f'(a+h).$$
They coincide in simple cases, but not in general. For example,
if
$$f(x)=\begin{cases}1,&x \ge 0 \\ 0,&x < 0\end{cases}$$
then $f'(x)=0$ for all $x\neq 0$, so $\lim_{x\to 0^\pm} f'(x)=0$,
but $f'(0)$ doesn't exist (since $f$ is discontinuous at $x=0$). More precisely, the right hand derivative $f'_+(0)$ is zero, but the left hand derivative ${f}'_{-}(0)$ is undefined.
A: Yes, it's possible! Consider the function
$$f(x) = x^2 W(x)$$
where $W$ is the Weierstrass function. At $x=0$ the derivative is $0$, but if it were differentiable anywhere else then $W$ would be differentiable too.
A: You ask about existence of a derivative in a single point, but in title you say continuity.
As for existence, a derivative $f'(a)$ of a real function $f(x)$ at point $x=a$ is defined as a limit
$$\lim_{h\to 0} \frac{f(a+h)-f(a)}h$$
The existence (and the value) of the limit determines a derivative at the chosen point, independent on the existence of the limit in any neighborhood of $a$. As others show, there exist functions which are differentiable at a single point only.
However if you ask for continuity, it requires the derivative to be defined (exist) in some neighbourhood of $a$, so that a limit of the derivative exists:
$$\lim_{x\to a}f'(x)$$
Then you can ask if a derivative is continuous at $a$. And there are functions (examples given in other answers) with a derivative discontinuous at some point, although existing in a neighborhood of that point.
