Null content is preserved by continuous functions, while null measure is not

Question

Given $X \subset R$ we say that $X$ has no content (or null content) and we denote it by $c(X) = 0 $ if given any $\varepsilon > 0$, there exist a finite collection of intervals $\{ I_k \}$ such that $X \subset \bigcup\limits_{k = 1}^n I_k$ and $\sum |I_k| < \varepsilon$, where clearly $|I_k|$ denotes the length of the interval. Now we say that $X$ has measure $0$, we accept any countable collection of intervals, under the same hypothesis.

Prove that the null content is preserved by continuous functions $f: \mathbb R \to \mathbb R$, and give an example of a continuous function $f$ such that $f(K)$ has nonzero measure, where $K$ denotes the cantor third set.

I don't know how to do it.

Answer 1

I'm sorry, I cannot post this as a comment since I don't have enough points, but, the function that takes the Cantor set to the unit interval is continuous, and, by Tietze extension, could be extended to a continuous function f: $\mathbb R\rightarrow \mathbb R$. The Cantor set itself has zero content, but its image is the unit interval, which does not.

BTW: I used to have an account here, but my computer crashed, and I lost the login info.