# Composition of Two Continuous Functions ... Browder, Proposition 3.12 ... ...

#### Peter

##### Well-known member
MHB Site Helper
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.12 ...

Proposition 3.12 and its proof read as follows:

In the above proof by Browder we read the following:

" ... ... Since $$\displaystyle f(I) \subset J$$, $$\displaystyle f^{ -1 } ( g^{ -1 }(V) ) = f^{ -1 } (U) \cap f^{ -1 } (J) = f^{ -1 } (U)$$ ... ... "

My question is as follows:

Can someone please explain exactly why/how $$\displaystyle f^{ -1 } (U) \cap f^{ -1 } (J) = f^{ -1 } (U)$$ ... ...

Help will be much appreciated ...

Peter