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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.

$${show}\:{that}\:{if}\:{f}\:{is}\:{a}\:{differentiable}\:{function}\:{at}\:{the}\:{point}\:{x}={a},\:{then}\:{f}\:{is}\:{continuous}\:{at}\:{x}={a}. \\ $$

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

if lim_(x→a) f(x)≠f(a) ⇒lim_(x→a) f(x)−f(a)≠0⇒  then f′(a)=lim_(x→a) ((f(x)−f(a))/(x−a))=+∞∨−∞  that is a contradiction.

$${if}\:{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)\neq{f}\left({a}\right)\:\Rightarrow{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)−{f}\left({a}\right)\neq\mathrm{0}\Rightarrow \\ $$$${then}\:{f}'\left({a}\right)={lim}_{{x}\rightarrow{a}} \frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}=+\infty\vee−\infty \\ $$$${that}\:{is}\:{a}\:{contradiction}. \\ $$

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

lim_(x→a) f(x)−f(a)=lim_(x→a) ((f(x)−f(a))/(x−a)).(x−a)=  lim_(x→a) ((f(x)−f(a))/(x−a)).lim_(x→a) (x−a)=  f′(a).0=0⇒lim_(x→a) f(x)=f(a)

$${lim}_{{x}\rightarrow{a}} {f}\left({x}\right)−{f}\left({a}\right)={lim}_{{x}\rightarrow{a}} \frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}.\left({x}−{a}\right)= \\ $$$${lim}_{{x}\rightarrow{a}} \frac{{f}\left({x}\right)−{f}\left({a}\right)}{{x}−{a}}.{lim}_{{x}\rightarrow{a}} \left({x}−{a}\right)= \\ $$$${f}'\left({a}\right).\mathrm{0}=\mathrm{0}\Rightarrow{lim}_{{x}\rightarrow{a}} {f}\left({x}\right)={f}\left({a}\right) \\ $$

Commented by psyche last updated on 24/Oct/19

thanks

$${thanks} \\ $$

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