Question Number 71769 by psyche last updated on 19/Oct/19
$${show}\:{that}\:{if}\:{f}\:{is}\:{a}\:{differentiable}\:{function}\:{at}\:{the}\:{point}\:{x}={a},\:{then}\:{f}\:{is}\:{continuous}\:{at}\:{x}={a}. \\ $$
Commented by kaivan.ahmadi last updated on 19/Oct/19
$${if}\:{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)\neq{f}\left({a}\right)\:\Rightarrow{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)−{f}\left({a}\right)\neq\mathrm{0}\Rightarrow \\ $$$${then}\:{f}'\left({a}\right)={lim}_{{x}\rightarrow{a}} \frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}=+\infty\vee−\infty \\ $$$${that}\:{is}\:{a}\:{contradiction}. \\ $$
Commented by kaivan.ahmadi last updated on 19/Oct/19
$${lim}_{{x}\rightarrow{a}} {f}\left({x}\right)−{f}\left({a}\right)={lim}_{{x}\rightarrow{a}} \frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}.\left({x}−{a}\right)= \\ $$$${lim}_{{x}\rightarrow{a}} \frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}.{lim}_{{x}\rightarrow{a}} \left({x}−{a}\right)= \\ $$$${f}'\left({a}\right).\mathrm{0}=\mathrm{0}\Rightarrow{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)={f}\left({a}\right) \\ $$
Commented by psyche last updated on 24/Oct/19
$${thanks} \\ $$