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**HallsofIvy** The derivative, at x= a, of f(x) is defined as

$\displaystyle \lim_{h\to 0}\frac{f(a+h)- f(a)}{h}$

Further, $\displaystyle \lim_{x\to a} g(x)= L$ if and only if $\displaystyle \lim_{n\to \infty} g(a_n)= L$ for every sequence $\displaystyle \{a_n\}$ converging to a.

Now, if there exist two sequences, $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$, converging to a, such that $\displaystyle \lim_{n\to\infty}\frac{f(a_n)- f(a)}{a_n-a}\ne \lim_{n\to\infty}\frac{f(b_n)- f(a)}{b_n- a}$ then the function is NOT differentiable at a. That is, if the two sequences have different limits, then the limit as h goes to 0 cannot exist and so the function is differentiable. For any a other than 0, take $\displaystyle \{a_n\}$ to be a sequence of rational numbers converging to a, and take $\displaystyle \{b_n\}$ to be a sequence of irrational numbers converging to a.