Continuity problems for my Analysis class

[LauraLovies]

I am having a lot of difficulty on my continuity problems for my Analysis class.

1. Prove that (f O g)(x) = f(g(x)) is continuous at any point p in R in three ways a.) Using the episolon delta definition of continuity, b.) using the sequence definition of continuity, and c.) using the open set definition of continuity.

2. Prove that if U is an open set in R, then its inverse is open.

 

[tiny-tim]

Hi LauraLovies!

(I assume f and g are both continuous? and have a delta: δ and an epsilon: ε )

Show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[LauraLovies]

f and g are both continuous so i know that there exists some $$\epsilon > 0 and greater than zero that fulfills the continuity definition. It just seems to obvious to me that i don't even know where to start. \delta$$

 

[tiny-tim]

f and g are both continuous so i know that there exists some \epsilon > 0 and greater than zero that fulfills the continuity definition. It just seems to obvious to me that i don't even know where to start. \delta

Start with an ε_f and δ_f, and an ε_g and δ_g, and then construct a proof using a new ε and δ based on them.

 

[lurflurf]

Which sequence definition? that lim f(x_n)=f(lim x_m)?
The general idea adaptible to the cases is that we desire
f(g(x+h))-f(x)=f(g(x)+[g(x+h)-g(x)])-f(x)
be small when h is