show-that-if-f-is-a-differentiable-function-at-the-point-x-a-then-f-is-continuous-at-x-a-

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

s h o w t h a t i f f i s a d i f f e r e n t i a b l e f u n c t i o n a t t h e p o i n t x = a, t h e n f i s c o n t i n u o u s a t x = a .

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

i f {l i m}_{x \to a} f (x) \neq f (a) \Rightarrow {l i m}_{x \to a} f (x) - f (a) \neq 0 \Rightarrow

t h e n f^{'} (a) = {l i m}_{x \to a} \frac{f (x) - f (a)}{x - a} = + \infty \lor - \infty

t h a t i s a c o n t r a d i c t i o n .

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

{l i m}_{x \to a} f (x) - f (a) = {l i m}_{x \to a} \frac{f (x) - f (a)}{x - a} . (x - a) =

{l i m}_{x \to a} \frac{f (x) - f (a)}{x - a} . {l i m}_{x \to a} (x - a) =

f^{'} (a) . 0 = 0 \Rightarrow {l i m}_{x \to a} f (x) = f (a)

Commented by psyche last updated on 24/Oct/19

t h a n k s

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