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Proof that Arcsin x is continuous ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
Can someone please help me to prove that the function f(x) = Arcsin x is continuous on the interval [-1, 1] ...

Peter
 

Cbarker1

Active member
Jan 8, 2013
236
What is the definition of continuous function in general?
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
An invertible function, 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].
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
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:



Abbott - Defintion 4.3.1 ... Continuity ... .png



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



Abbott - Theorem 4.3.2 ... Characterizations of Continuity ... .png


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

- - - Updated - - -

An invertible function, 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