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- Jun 22, 2012

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Peter

- Thread starter Peter
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- Thread starter
- #1

- Jun 22, 2012

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Peter

- Jan 29, 2012

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- Jun 22, 2012

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What is the definition of continuous function in general?

The definition of continuity in \(\displaystyle \mathbb{R}\) is given in Stephen Abbott's book: Understanding Analysis, as follows:

Alternative characterizations of continuity are given by Abbott in Theorem 4.3.2 as follows:

So to show (from first principles) that \(\displaystyle \text{Arcsin } x\) is continuous on \(\displaystyle [-1, 1]\) we would have to show that given an arbitrary point \(\displaystyle c \in [-1, 1]\) that for every \(\displaystyle \epsilon \gt 0\) we can find \(\displaystyle \delta \gt 0\) such that

\(\displaystyle \mid x - c \mid \lt \delta \ \Longrightarrow \ \mid \text{Arcsin x } - \text{Arcsin } c \mid \lt \epsilon\) ...

But how do we proceed ... ?

Peter

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invertiblefunction, y= f(x), is continuous at [tex]x= x_0[/tex] if and only if [tex]y= f^{-1}(x)[/tex] is continuous at [tex]x= f(x_0)[/tex].

Thanks for the help, HallsofIvy ...

Peter