I believe that I may have once seen a definition of the derivative that went along these lines:


Here, $(x_n)$ is a sequence that converges to $c$.

As I do not recall much about it, I may have left out one or two important assumptions. However, does this seem in any way correct? And if so, does this theorem/etc. have a name? Thanks in advance!

  • 4
    $\begingroup$ If $f$ is differentiable at $c$, then this will work. In order to make this a definition, you want to say that $f$ is differentiable at $c$ with derivative $D$ (a number) if and only if for every sequence $(x_n)$ that converges to $c$ we have $\lim\limits_{n\to\infty}\frac{f(x_n)-f(c)}{x_n-c} = D$. This would be analogous to the definition of sequential continuity, which says that $f$ is continuous at $a$ if and only if for every sequence $x_n$ that converges to $a$, we have $\lim_{n\to\infty}f(x_n) = f(a)$. $\endgroup$ Mar 21, 2012 at 4:13

1 Answer 1


One might say that the derivative $f'(x)$ exists at $c$ if and only if $\lim \dfrac{f(x_n) - f(c)}{x_n - c}$ exists and is the same for any sequence $x_n$ s.t. $x_n \to c$

If you parse what this means, you can see that this is equivalent to the standard definition. In particular, it's easy to see that if a function isn't differentiable under this definition, then it's not in the standard, and vice versa.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .