We show that, with the exception of the symmetric derivative, each limit of the form

$$\begin{aligned} \lim _{h\rightarrow 0}\frac{Af(x+ah)+Bf(x+bh)}{h},\qquad (A+B=0,Aa+Bb=1), \end{aligned}$$

is equivalent to the ordinary derivative, for all continuous functions at x. And, up to a non-zero scalar multiple, these are the only criteria for differentiating all continuous functions at x, by taking limits of first order difference quotients with two function evaluations.

中文翻译：

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连续函数的微分准则

我们证明，除了对称导数之外，形式的每个极限

$$ \ begin {aligned} \ lim _ {h \ rightarrow 0} \ frac {Af（x + ah）+ Bf（x + bh）} {h}，\ qquad（A + B = 0，Aa + Bb = 1），\ end {aligned} $$

对于x处的所有连续函数，等价于普通导数。而且，对于非零标量倍数，这是通过对带有两个函数求值的一阶差分商的限制来区分x处所有连续函数的唯一标准。