Short and Sweet Calculus

3.2 Differentiability and Continuity

The following theorem says that differentiability implies continuity \[\boxed{\text{Differentiability}\Rightarrow\text{Continuity}}\]

3.1. If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

Proof. If \(x\neq a\), we can write \[f(x)=\frac{f(x)-f(a)}{x-a}(x-a)+f(a).\] Then by the limit rules \[\begin{aligned} \lim_{x\to a}f(x) & =\lim_{x\to a}\left[\frac{f(x)-f(a)}{x-a}(x-a)\right]+f(a)\\ & =\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\lim_{x\to a}(x-a)+f(a)\\ & =f'(a)\cdot0+f(a)=f(a).\end{aligned}\] ◻

It follows from the above theorem that if a function is not continuous at a point, it is not differentiable there.

\[\boxed{\text{Not Continuous}\Rightarrow\text{Not Differentiable}}\]

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