Differentiable function

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A function f(x) is differentiable at the point a if and only if, as x approaches a (which it is never allowed to reach), the value of the quotient:

\frac{f(x) - f(a)}{(x - a)}

approaches a limiting value that we call the derivative of the function f(x) at x=a.

There is also the more rigorous ε − δ definition: a function f is said to be differntiable at point a if ∀ε > 0 ∃​δ > 0 such that if

 |x - a| < \delta\,


|\frac{f(x) - f(a)}{x-a} - f'(a) |  < \epsilon \,.
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