Can an interval with all positive functions of x be proven?

[mathkillsalot]

Can an interval with all positive functions of x be proven??

f(x) has to always be greater that 0. How do you prove this??

f is continuous as c and f(c)>0. prove that there is an open interval (a.b) centered at c such that f(x)>0 for all x that are elements of (a,b)

 

[gb7nash]

This comes straight from the definition of continuity. Apply the definition of continuity to your point c. What does it say?

 

[mathkillsalot]

lim f(x) = f(c)
x -> c ?

but i don't get how it proves that f(a) and f(b) are positive...

 

[gb7nash]

There's another definition of continuity. Use the epsilon-delta definition and choose an appropriate value for epsilon.

 

[mathkillsalot]

wait...so since f(c) is positive, and (a,b) are approaching c, it means that f(a) and f(b) are approaching a positive number f(c)? and that they are positive?

 

[gb7nash]

wait...so since f(c) is positive, and (a,b) are approaching c, it means that f(a) and f(b) are approaching a positive number f(c)? and that they are positive?

I'm not sure what you mean by (a,b) approaching c, but it sounds like you do have the right idea. Since f is continuous at c, we can choose values that are close enough to c (a and b) such that f(a) and f(b) are both positive and f is positive on (a,b). Unfortunately, your proof needs to be a little more rigorous than this. From here, you need to use the epsilon-delta definition of continuity to actually prove it.

 

[mathkillsalot]

okaaay...thankyou
and I forgot something, we're supposed to apply the sign preserving property on this. Do i still need the epsilon-delta definition of continuity?
can't I just assign positive values for x?? or is that wrong?

 

[gb7nash]

If you can use the sign-preserving property, it's trivial. The answer follows straight from the definition and you don't need the e/d definition.

 

[mathkillsalot]

okay, thank you so much )))))

 

[HallsofIvy]

Essentially the same question (about proving the "sign preserving property") was asked and answered here-
https://www.physicsforums.com/showthread.php?t=460196

By the way, "mathkillsalot" you might want to change that user-name. The very people who would help you the most are likely to take exception to it.

 

[mathkillsalot]

thankyou
and I just requested (chroot?) to change my username... hahaha