Continuous image of set having finite Lebesgue outer measure .

Question

Let $f:\Bbb R\rightarrow\Bbb R$ be a continuous function and $E\subseteq\Bbb R$ having finite outer measure. Then can I say that image of $E$ also has finite outer measure ? If I take linear maps it works . In general I have no idea . Please suggest. Thanks.

Answer 1

Let $f:\Bbb R\rightarrow \Bbb R$ be the continuous function defined as $$f(x)=e^x$$

Consider the set $$E=\cup_{n=1}^\infty E_n, E_n=[n^2,n^2+\frac{1}{n^2}]$$ Then $$m(E)=\sum\frac{1}{n^2}<\infty$$ but $$m(f(A))=\sum e^{n^2}(e^{\frac{1}{n^2}}-1)=\infty.$$