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

$$f'(c)=\lim_{n\to\infty}\frac{f(x_n)-f(c)}{x_n-c}$$

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!

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    $\begingroup$ 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

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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.

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