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One exercise from a list. I have no idea how to finish it.

Let $I=[c,d]\subset \mathbb{R}$.

Let $f:I\to \mathbb{R}$ be continuous at $a\in (c,d)$.

Suppose that there exists $L\in \mathbb{R}$ such that $$\lim \frac{f(y_n)-f(x_n)}{y_n-x_n}=L$$ for every pair of sequences $(x_n),(y_n)$ in $I$, with $x_n and $\lim x_n=\lim y_n=a$.

Prove that $f'(a)$ exists and it is equal to $L$.

Any help? Thanks in advance.

  • 2
    "I have no idea how to finish it." So I guess you started somewhere? What's your idea?2012-08-20
  • 1
    I tried to consider $z_n=y_n-x_n$ which converge to zero.2012-08-20
  • 0
    I have no idea what *I tried to consider $z_n=y_n−x_n$* could mean. You might want to explain.2012-08-20
  • 0
    Can you just do $\forall n\in\mathbb{N}, x_n=a$?2012-08-20
  • 1
    I think they require $x_n < a$2012-08-20
  • 0
    Could you first prove that $(f(y_m)-f(x_n))/(y_m-x_n)\to L$ as both $m,n\to\infty$, for any such pair of sequences $(x_n),(y_n)$?2012-08-20
  • 0
    @SeanEberhard, this is the hypothesis.2012-08-20
  • 0
    @Sigur Not quite. It's very slightly but very usefully different, since you could then let $x_n\to a$ first (using continuity of $f$).2012-08-20
  • 0
    @SeanEberhard, your suggestion is to consider sequences with different indexes, but still with the inequality?2012-08-20
  • 0
    @Sigur Yes. That's what I would try.2012-08-20

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