TITLE: Continuous versus differentiable
QUESTION [25 upvotes]: A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it.
Until today, I thought these were merely two equivalent definitions of the same concept. But I've read some stuff today which seems to be claiming that this is not the case.
The obvious next question is "why?" Apparently somebody has already asked:
Are Continuous Functions Always Differentiable?
Several answers were given, but I don't understand any of them. In particular, Wikipedia and one of the replies above both claim that $|x|$ has no derivative. Can anyone explain this extremely unexpected result?
Edit: Apparently some people dislike the fact that this is non-obvious to me. To be clear: I am not saying that the result is untrue. (I'm sure many great mathematicians have analysed the question very carefuly and are quite sure of the answer.) I am saying that it is extremely perplexing. (As a general rule, mathematics has a habit of doing that. Which is one of the reasons why we demand proof of everything.)
In particular, can anyone explain precisely why the derivative of $|x|$ at zero is not simply zero? After all, the function is neither increasing nor decreasing, which ought to mean the derivative is zero. Alternatively, the expression
$$\frac{|x + a| - |x - a|}{a}$$
becomes closer and closer to zero as $a$ becomes closer to zero when $x=0$. (In fact, it is exactly zero for all $a$!) Is that not how derivatives work?
Several answers have suggested that the derivative is not defined here "because there would be a jump in the derivative at that point". This seems to assert that a continuous function must never have a discontinuous derivative; I'm not convinced that this is the case. Can anyone confirm or refuse this argument?

REPLY [2 votes]: So, the answer to this question really depends on your notion of differentiability. Let us start with the classical notion of differentiability. A function, $f(x)$ is differentiable at a point, $x_0$ if the following limit exists:$lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$. The other answers give a good explanation of why $|x|$ is not differentiable (in the classical way). 
That said, their are other notions of differentiability that one may contemplate! For example the function that you mentioned $f(x)=|x|$, one may assign a derivative of 0 when $x=0$, if we use good generalization of derivative. One way to do this is to see that the derivative of $|x|$ is $+/-1$ depending on whether $x$ is greater than or less than zero. But what happens at zero. Well if you write this derivative function (that is undefined at 0 for the moment) as a Fouier series, then evaluate the series at $x=0$, you will get that the "derivative" that you obtain is zero (I am sure their are better ways of doing this such as approximating the +/- function by smooth functions).
That said, if we are speaking about non-classical notions of differentiability, one may even differentiate discontinuous functions in the distributional way.  For instance the "second derivative" again non-classical of $|x|$ is the dirac delta distribution (which is no longer a real valued function but a certain type of limit of real valued functions). Also the derivative mentioned of $|x|$ is also of distributional type.
The wikipedia artical on distributional derivatives.
http://en.wikipedia.org/wiki/Distribution_%28mathematics%29