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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:

View attachment 9519

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

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:

View attachment 9519

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