Differentiability of a Function

==2024-12-12

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Format your brain.

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Theory

A function is differentiable at an interval [a,b] if:

$$\underset{x\to a^{+}}{\lim }\frac{f(x)-f(a)}{x-a}\text{ is defined.}$$

$$\underset{x\to b^{-}}{\lim }\frac{f(x)-f(b)}{x-b}\text{ is defined.}$$

3. Both LHL and RHL are defined and equal.

$$\forall c\in [a,b],\underset{x\to c^{+}}{\lim }\frac{f(x)-f(c)}{x-c}=\underset{x\to c^{-}}{\lim }\frac{f(x)-f(c)}{x-c}$$

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PTR