# Proof that Arcsin x is continuous ...

#### Peter

##### Well-known member
MHB Site Helper
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
What is the definition of continuous function in general?

#### HallsofIvy

##### Well-known member
MHB Math Helper
An invertible function, y= f(x), is continuous at $$x= x_0$$ if and only if $$y= f^{-1}(x)$$ is continuous at $$x= f(x_0)$$.

#### Peter

##### Well-known member
MHB Site Helper
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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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