What technique should I apply to find the derivative of a ceiling or floor function e.g d/dx(x*⌈x⌉) and d/dx(x*⌊x⌋)? May I know the technique to apply to find the derivative, whenever I see a ceiling function of floor function. Thank You!
e.g   $$ \frac{d}{dx}(x*\lceil x \rceil )$$  and $$ \frac{d}{dx}(x*\lfloor x \rfloor )$$ 
Is there a solution to $ \frac{d}{dx}(x*\lceil x \rceil )$ ?
 A: Usually for a ceiling function it can be written as function by cases. For example, if  $f(x)=x* \lceil x \rceil $, then taking $f$ defined on the interval  $]0,5[$, the  $f$  can be written as $$ f(x)=\begin{cases}
 x   \quad \text{ if } \quad 0<x \leq 1  \\ 
 2x  \quad \text{ if } \quad 1<x \leq 2  \\ 
 3x  \quad \text{ if } \quad 2<x\leq 3  \\ 
 4x  \quad \text{ if } \quad 3<x\leq 4  \\ 
 5x  \quad \text{ if } \quad 4<x< 5  \\ 
\end{cases} $$
And as we can see, $f$ is even not continuous  at  $1,2,3,4$ , ingeneral such $f$ will not be conyinuous at any inetger even if it is defined on  $\mathbb{R}$. Hence,  it is not differentiable at any inetger, and thus no derivative exists at any  $x=n$ with  $n$ integer. On the other hand, if  $x$ is not an integer, as we can see from the above example we can deal with  $f$ depending its value in each open interval  $]n,n+1[$. So for the above example we can see  
$$ f'(x)=\begin{cases}
 1   \quad \text{ if } \quad 0<x <1  \\ 
 2  \quad \text{ if } \quad 1<x <2  \\ 
 3 \quad \text{ if } \quad 2<x< 3  \\ 
 4  \quad \text{ if } \quad 3<x< 4  \\ 
 5  \quad \text{ if } \quad 4<x< 5  \\ 
\end{cases} $$
So  $f'(x)=\lceil x \rceil $ if $x$ is not an integer, and  $f'(x)$ is not defined  if  $x$ is an integer.
Anologously we may treat the case of  the floor function as  the case of the  ceil function.
