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.