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Loser66
 one year ago
If f(x) is differentiable at x0. Prove it continuous at x0
Please, help
Loser66
 one year ago
If f(x) is differentiable at x0. Prove it continuous at x0 Please, help

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Loser66
 one year ago
Best ResponseYou've already chosen the best response.0@xapproachesinfinity hey kid, help old man please

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1hey old man long time

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1i think theorem state if f is differentiable applies f is continuous so you are proving the theorem

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1so let see f differentiable at x0 means the limits exist f'(x0)

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1so \(\lim_{x\to x_0} \frac{f(x)f(x_0)}{xx_0}=\text{exist}\)

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1you have to work from here

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1first let's state the definition of f being cont at x0 that is to say \(\lim_{x\to x_0} f(x)=f(x_0)\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0Sorry, kid, old man was kicked out of the site.

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0But I think you got circulation argument: differentiable > continuous 

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0However, I got it. \(f(x)  f(x_0) = \dfrac{f(x) f(x_0)}{xx_0}(xx_0)\) \(lim_{x\rightarrow x_0} f(x)  f(x_0) = lim_{x\rightarrow x_0}\dfrac{f(x) f(x_0)}{xx_0}xx_0\) since f(x) differentiable at \(x_0\), the limit exists and as x approaches \(x_0\) \(xx_0=0\) That gives us \(lim_{x\rightarrow x_0}f(x) f(x_0)=0\) or \(lim_{x\rightarrow x_0}f(x) = f(x_0)\) That shows f(x) continuous at x0

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1there is no circulation it is one way f different implies f continuous so you start with f different stating the definition limf(x)f(x0)/xx0=f'(x0) multilply both sides by (xx0) we get lim(f(x)f(x0)=f'(x0)(xx0) as x tends to x0 the quanties (xx0) shrinks to zero (limit concept) then we are safe to say lim(f(x)f(x0)=f'(x0)0=0 therefore lim(f(x))=f(x0) hence f is continuous at x0

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.1a subtlety here is the xc is zero but it is not zero however we write as it is zero
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