Clarifying the statement 'a function has derivative' In high-school I used to think that either a function has a derivative or doesn't. But then in college everything becomes subtler, I encounter example where a function can be differentiable on some interval but not on the other. 
Say $|x|$ is differentiable on $(0, 1)$ but not on $[-1,1]$ because of the problem with $x=0$. My question is, when someone say $f(x)$ has derivative or is differentiable, only that, then to which interval he is actually referring in general? Is it simply the whole domain on which $f(x)$ is defined? In the case of $|x|$ I understand it is $\mathbb{R}$. 
Specifically, If I show that this limit exist, have I also shown that a function is differentiable anywhere it is defined? 
$$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
Lastly, is it sufficient then just to show one point that doesn't have the same limit in order to show that a function has no derivative? 
Thank You
 A: Differentiability is defined at each point seperately: the function is differentiable at the point $x$ if the limit of $\frac{f(x+h)-f(x)}{h}$ as $h\to0$ exists. If you say that $f$ is differentiable on a set $A$, it means it is differentiable at each $x\in A$. One might just say that "$f$ is differentiable" if it is differentiable at every point in its domain.
A: There is no standard meaning to the phrase "a function has derivative", unless such a concept is defined in your textbook or course.
It can mean existence of a derivative in a set that is clear from context.
It can mean that the derivative is also continuous in this set.
It can mean that the derivative exists at a single point.
(If you carry on studying analysis, it can also refer to something called the weak derivative.)
You should try to be (and ask others to be) more specific.
That is, say "$f$ has a derivative on $(0,1)$", "$f$ is continuously differentiable on $(0,1)$" or "$f$ has a derivative at $\frac12$".
You can replace "has derivative" with "is differentiable", of course.
Typically differentiability means continuous differentiability in the whole domain, but you should be careful with vague statements.
For example, a function can be differentiable everywhere without it derivative being everywhere continuous.
Also remember that differentiability is defined pointwise, as Holographer pointed out.
