This question was asked to me by brother and I am not sure about its answer. So, I am asking it here.
As we know that a function $f :U $ (subset of $\mathbb{R}$) $ \to\mathbb{R} $ , defined on an open set $U$ is differentiable at $a\in U$ if
the derivative $\lim_{h\to 0}\frac{ f(x+h)- f(x) } { h }$ exists.
My question is can we change the definition of differentiability to $\lim_{h\to 0}\frac{ f(x+h) – f(x-h) } {h} $ exists.
Why or Why not?
Best Answer
Well of course you can change it. Definitions can be whatever we want then to be. I think you mean are the 2 definitions any different. Yes they are different. The latter definition is symmetric about h and -h, meaning it doesn't matter if you take the limit from the right or the left, the result is the same, which means every point is differentiable. Even corners like y = abs(x) at the origin (the derivative is 0). A geometric interpretation is what you are doing is taking the limit as h goes to 0+ and the limit as h goes to 0- and computing the average between the two. For continuous functions, by definition the limits are the same, so you just get the normal derivative. But for non-continuous functions like y = abs(x), the normal derivative isn't defined at 0, because it's -1 coming from the left and +1 coming from the right. So this new definition says well let's just take the average of -1 and 1 which is 0 and say that's the derivative.
Nice extension of the derivative but doesn't have many further uses I think...