Show that arctan(x) exists and is continuous

[Kate2010]

Homework Statement

Show tan(x): (-pi/2, pi/2) -> R has a continuous inverse arctan(x) : R -> (-pi/2, pi/2).

You may assume that tan(x) is continuous and strictly increasing on the given domain, and tends to +/- $$\infty$$ at +/- pi/2

Homework Equations

The Attempt at a Solution

I think I have shown that tan(x) has an inverse on this domain by showing it is bijective.

However, I am unsure how to go about showing that arctan(x) is continuous.

 

[vela]

You could try considering tan(arctan(x))=x and show that if arctan(x) is not continuous, it leads to a contradiction.

(I'm just offering a suggestion. I think it'll work, but I didn't actually try it.)

 

[Hurkyl]

Continuous functions can have discontinuous inverses.

I'm not sure if this is possible when the domain and codomain are intervals, though. I suspect not.

 

[boboYO]

What characterizations of continuity have you learnt? Try showing that tan takes every open set to an open set.