
Zero content
A set z in R is said to have zero content if for any epsilon > 0, there is a finite collection of intervals I1, ...IL such that z is a subset of the union of those intervals (meaning all points in z are contained in the union of all the intervals) AND the sum of the lengths of the intervals is less than epsilon.
Thus, I was wondering if I could clarify something about zero content. So, for example, the set z = {y : y = x^2, x is part of [0, 10]} would not have zero content because for all the points z to be contained in intervals, the length of union of the intervals must be 10 and epsilon can be found which is less than 10.
However, am I right in thinking that the set z = {y : y = x^2, x is part of integers} has zero content? This is because arbitarily small intervals can be chosen around those points, and the union of the intervals can be made less than epsilon.
So basically, if f is continuous function, f(S) has zero content only when S is disconnected? Like sequences mapped by continuous functions have zero content because they are indexed by integers which form a disconnected set?

If $\displaystyle f$ is not constant then yes, because if $\displaystyle f(s)$ has two elements say $\displaystyle x<y$ then, since $\displaystyle f$ is continous and $\displaystyle S$ is connected then $\displaystyle f(S)$ is connected which means $\displaystyle (x,y) \subset f(S)$ which means the content of $\displaystyle f(S)>0$.