Proving Continuity with Epsilon-Delta: How to Approach a Challenging Function?

[tonix]

Hi

I am trying to prove the continuity of a function. I do understand the definition and I can do it for "smaller" functions. However, for this "larger" function I am having troubling bounding it and thus can't find a prove. Any suggestions would be greatly appreciated!

Show, using the epsilon-delta definition, that the following function is continuous: f(y) = 1 / (y^4 + y^2 + 1).

 

[matt grime]

Wow, you're really expected to do that? Try looking at the proofs that if f is continuous at a point and not zero there that 1/f is continuous there to see how to do it for this particular example.

 

[matt grime]

Actually, scrub that, you can do it without too much difficulty, in a manner of speaking.

Suppose |u|<|v|, and |u-v| <d, and that d is chosen such that |v|<2|u|.

then |f(u)-f(v)| = |u-v||g(u,v)| where g(u,v) you can work out after simplification is a fraction with top and bottom some polynomials in u and v. the bottom is striclty larger than 1, so the whole thing is in abs value less than:

d|u^3+u^2v+uv^2+v^3+u+v|, we may bound all this by putting in 2|u|, and picking d such that...

 

[tonix]

Thank you. That helped.