Q:

measure zero point in differentiable "almost everywhere"

I am a new student in leaning real analysis and still confused about "almost everywhere.
Definition:  

Points $x\in \mathcal{X}$ is true  except for $x$ in some null set (with measure $0$) ", we way it is true a.e.   

Example: 
"Differentiable almost everywhere" means  differentiable at every point outside a set of (Lebesgue) measure zero $\mu = 0$. 
My question is:  
Where is the set with measure zero in real line for example?  
Hope to construct correct concept. 

A:

Differentiability a.e. means that there is some null set $N$ such that your function (say) $f$ is differentiable on $ℝ \setminus N$. This null set depends on $f$. One example is
$$f(x) = |x|$$
This function has $N=\{0\}$.
Another is the Cantor function, and it is differentiable a.e. with derivative $0$, and is not differentiable on the Cantor set.
Yet another example is 
$$f(x) = 1$$
This function is differentiable a.e., in fact its differentiable everywhere but this doesn't stop you from setting $N=∅$.

