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Question Number 71769 by psyche last updated on 19/Oct/19

$$showthatiffisadifferentiablefunctionatthepointx=a,thenfiscontinuousatx=a.$$

Commented by kaivan.ahmadi last updated on 19/Oct/19

$$if{lim}_{x\to a}f\left(x\right)\ne f\left(a\right)\Rightarrow {lim}_{x\to a}f\left(x\right)-f\left(a\right)\ne 0\Rightarrow$$$$then{f}^{\prime }\left(a\right)={lim}_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-a}=+\mathrm{\infty }\vee -\mathrm{\infty }$$$$thatisacontradiction.$$

Commented by kaivan.ahmadi last updated on 19/Oct/19

$${lim}_{x\to a}f\left(x\right)-f\left(a\right)={lim}_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-a}.(x-a)=$$$${lim}_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-a}.{lim}_{x\to a}(x-a)=$$$$f^{\prime }\left(a\right).0=0\Rightarrow {lim}_{x\to a}f\left(x\right)=f\left(a\right)$$

Commented by psyche last updated on 24/Oct/19

$$thanks$$

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