I wish to show f(x)=x3 such that x is an element of the real numbers is continuous at an arbitrary x0.
This means that, given some epsilon greater than zero (henceforth referred to simply as "E"), there exists a delta greater than zero (henceforth referred to as "D") such that...
|x-x0|<D implies |f(x)-f(x0)|<E
So, we have...
|x-x0|<D should imply |x3-x03|<E.
This means...
|x-x0||x2+x0x+x02|<E
I see that some value less than delta shows up here, but don't know what to do from here. I see several other relations that are seemingly useless, such as...
|x|<D+|x0|
Where do I go from here? I know I need to figure out what to pick as my delta, but I don't know how to do this.