Is it true that differentiable functions can have essential discontinuity I was reading wiki and found this statement. Quoting it:

A function f is said to be continuously differentiable if the
  derivative f'(x) exists and is itself a continuous function. Though
  the derivative of a differentiable function never has a jump
  discontinuity, it is possible for the derivative to have an essential
  discontinuity.

I am confused with this statement as we know :
Function is differentiable in its domain -> continuous in that domain
Not continuous -> Not differentiable 
How does a function still be differentiable even if it is not continuous ? 
How It can have essential discontinuity ? I did not get it from wiki example. 
Please clarify :)
 A: The passage in question is saying that the derivative has an essential discontinuity, not the original function.  That is, it is possible to have a differentiable function $f$ such that the function $f'$ has an essential discontinuity.  Of course you are correct that if $f$ is differentiable, then $f$ must be continuous.
A: Here are some points which may be helpful for you.


*

*$f(x)$ is derivative $\to$ $f(x)$ is continuous; however the converse doesn't hold, for example, $y=|x|$ at $x=0$. 

*A function $f$ is said to be continuously differentiable: $f$ is derivative
and $f'(x)$ is continuous.
A: No. A necessary condition for $f(x)$ to be differentiable that it be continuous. 
It is straightforward to visualize if you have a function, say, $f(x) = \frac{1}{x}$, which is discontinuous at $x = 0$, and then augment it with $f(0) = k$ (for some Real $k$ of your choosing), you still have a discontinuous function at $x=0$ even though there is now an essential singularity at there. As such, $f(x)$ cannot be differentiable at $0$.     
