1. ## First fundamental theorem, question about continuity

Hi,

I was reading the proof for the first fundamental theorem in Apostol, and one part of it says the following.

Continuity of $\displaystyle f$ at $\displaystyle x$ tells us that, if $\displaystyle \epsilon$ is given, there is a positive $\displaystyle \delta$ such that

$\displaystyle |f(t)-f(x)|<\frac{1}{2}\epsilon$

whenever

$\displaystyle x-\delta < t < x+\delta$

I do not see how $\displaystyle |f(t)-f(x)|$ must be less than $\displaystyle \frac{1}{2}\epsilon$. I do see why it has to be less than $\displaystyle \epsilon$ though..

Any hints?

Thanks.

2. It's simply a matter of formalism - the author chose to take $\displaystyle \epsilon_0 = \frac{1}{2} \epsilon$ and used $\displaystyle \epsilon_0$ in the definition of continuity.

3. So he just chooses a smaller neighborhood?

4. Well, it doesn't really matter. The proof would've worked as well, had he taken $\displaystyle \epsilon$ instead of $\displaystyle \frac{1}{2} \epsilon$. The only difference would be that in the end, he would have $\displaystyle | \text{something} | < 2 \epsilon$, and by taking $\displaystyle \epsilon_0 = \frac{1}{2} \epsilon$, he will end up with $\displaystyle | \text{something} |< \epsilon$.

5. Ah, he's really thinking ahead then. Must be a good chess player
Thanks mate.

6. Originally Posted by Mollier
Ah, he's really thinking ahead then. Must be a good chess player
Thanks mate.
Actually, most proofs like this are developed by taking "$\displaystyle \epsilon$", seeing at the end that it would have been better to use "$\displaystyle \frac{1}{2}\epsilon$", then going back and changing it- which you can't do in chess.