Proof that Arcsin x is continuous ....

[Math Amateur]

Can someone please help me to prove that the function f(x) = Arcsin x is continuous on the interval [-1, 1] ...

Peter

 

[cbarker1]

What is the definition of continuous function in general?

 

[HallsofIvy]

An invertible function, y= f(x), is continuous at $$x= x_0$$ if and only if $$y= f^{-1}(x)$$ is continuous at $$x= f(x_0)$$.

 

[Math Amateur]

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:
View attachment 9348
Alternative characterizations of continuity are given by Abbott in Theorem 4.3.2 as follows:
View attachment 9349So 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 $$x= x_0$$ if and only if $$y= f^{-1}(x)$$ is continuous at $$x= f(x_0)$$.

Thanks for the help, HallsofIvy ...

Peter