[If f′(x0) exists, then in the neighborhood of x0, we have f(x0+Δx)−f(x0)≈f′(x0)Δx.]
[Because Δx = x − x0, we can get f(x) = f(x0) + f′(x0)(x − x0) + o(x − x0).]
[Thus, we can approximately write f(x) ≈ f(x0) + f′(x0)(x − x0)....]
This is a very fundamental differentiation formula, not much different from what Newton established in history.
But as he kept reading—
Xu Yun suddenly froze, his expression started becoming more serious:
"However, during the derivation, I suddenly discovered a problem."
"That is, the concepts of 'infinitesimal', 'approaching infinity', dx, all seem very vague—they are sometimes zero, sometimes not, which inevitably causes confusion."
"So I spent another two and a half years, and finally deduced a more rigorous mathematical concept."
"If and only if, for any ε, there exists a δlim0 such that as long as 0 & lim |x-a| & → δ, we have |f(x) − L| lim ε."
