Why we cant take average of slope for non differential points 
Why can't we use the formula of $$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} $$
it gives the same result for continuous functions like $x^2$ and for discontinuous function it gives average of the slope on either side.
It makes sense too as seen in the graph of velocity of ball as it touches the ground it should have $0$ velocity.
Also as derivative is the approx of the values around that point it makes sense that at the corner of a discontinous function the derivative is the average of the slope on either side.
so for $|x|$ the slope on either side of $0$ is $-1$ and $+1$ so the average should be $0$
 A: The "average derivative" formula you give is more commonly known as the symmetric derivative.
The problem is that certain results that are true for the standard version of the derivative do not hold if you replace it with the symmetric version. Probably the most important of these is the mean value theorem (MVT).
The MVT states that if $f$ is differentiable on the open interval $(a, b)$ and continuous on its closure $[a, b]$, then you can find a value $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$, i.e. the tangent at $c$ is parallel to the secant from $a$ to $b$. This result is used in a lot of proofs of important identities and theorems.
If we consider the function $f(x) = |x|$, then its derivative is $-1$ on $(-\infty, 0)$ and $1$ on $(0, \infty)$, and its symmetric derivative is $0$ at $x = 0$. But it's very easy to draw secants of many different gradients other than $0$ or $\pm 1$, which breaks the MVT.
Generally speaking, introducing something that is "nicer" because it works on a broader class of object will often break a nice property of the narrower class, and there will always be an even broader class of object that will give you problems. So it comes down to what is most useful for what you're trying to do - for modelling the velocity of an object maybe existence is more important than the MVT, but if you're trying to prove the validity of something relating to solutions of differential equations or the like then maybe you need the MVT so the symmetric derivative isn't appropriate.
