Hi I have a question about continuity as it is proven by the existence of a derivative.(adsbygoogle = window.adsbygoogle || []).push({});

The proof I've read is the following and I really just want to talk about it to be 100% sure I've understood it and I know where it comes from;

1:We'll take the equation of a line; [tex] f(x) \ - \ f(x_0) \ = \ m (x \ - \ x_0) \ [/tex]

where;

[tex] m \ = \ \frac{f(x) \ - \ f(x_0)}{x \ - \ x_0} [/tex]

and set it up like so;

2:[tex] f(x) \ = \ \frac{f(x) \ - \ f(x_0)}{x \ - \ x_0} (x \ - \ x_0 ) \ + \ f(x) [/tex]

So, as I understand the proof, if we want to prove that the function f is continuous we will take the limit of the above expression and as x tends back onto itself and by getting the equality f(x_0) = f(x_0) we've proved the function continues along at this point

3:[tex] \lim_{x \to x_0} f(x) \ = \ \lim_{x \to x_0} \frac{f(x) \ - \ f(x_0)}{x \ - \ x_0} (x \ - \ x_0 ) \ + \ \lim_{x \to x_0} f(x) [/tex]

4:[tex] f(x_0) \ = \ f ' (x_0) (x_0 \ - \ x_0 ) \ + \ f(x_0) [/tex]

5:[tex] f(x_0) \ = \ f ' (x_0) (0) \ + \ f(x_0) [/tex]

6:[tex] f(x_0) \ = \ f(x_0) [/tex]

I wonder how deep a proof this is, it is from an advanced calculus text. :tongue2:

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# Definition of Continuity?

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