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Question Number 2219 by tabrez8590@gmail last updated on 09/Nov/15

i t i s n e c e s s a r y f o r d i f f r e n t i a b l e f u n c t i o n i t m u s t b e c o n t i n u t y w h y ? a n d h o w p l e a s e e x p l a i n b y e x a m p l e

Answered by Filup last updated on 09/Nov/15

if {\begin{cases} y = x if x > 0 \\ y = - x if x < 0 \end{cases}

this function is differentiable at all

points except x = 0

Hense this function is not continuous

Answered by prakash jain last updated on 09/Nov/15

f (x) - f (x_{0}) = \frac{f (x) - f (x_{0})}{x - x_{0}} \times (x - x_{0})

\lim_{x \to x_{0}} [f (x) - f (x_{0})] = \lim_{x \to x_{0}} \frac{f (x) - f (x_{0})}{x - x_{0}} \times \lim_{x \to x_{0}} (x - x_{0})

If f (x) is differentiable at x = x_{0} . Then

\lim_{x \to x_{0}} \frac{f (x) - f (x_{0})}{x - x_{0}} = f^{'} (x_{0}) exits .

\lim_{x \to x_{0}} [f (x) - f (x_{0})] = f^{'} (x_{0}) \times 0 = 0

o r \lim_{x \to x_{0}} f (x) = f (x_{0})

Thus f (x) is continuous at x = x_{0} by definition

of continuity .

If f (x) is differentiable at x = x_{0} then it is

continuous a x = x_{0} .