I wish to show f(x)=x^{3}such that x is an element of the real numbers is continuous at an arbitrary x_{0}.

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-x_{0}|<D implies |f(x)-f(x_{0})|<E

So, we have...

|x-x_{0}|<D should imply |x^{3}-x_{0}^{3}|<E.

This means...

|x-x_{0}||x^{2}+x_{0}x+x_{0}^{2}|<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+|x_{0}|

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.