# Which functions map dense subsets onto subsets?

I was wondering about the following:

Let $X$ be a normed space and $A$ be a dense subset in $X$. What properties must a function $f$ necessarily have, so that we have that $f(A)$ is dense in $Y$, if we have $f:X \rightarrow Y$. I think it is pretty clear that the function should be bijective? Do we need more(continuous?)

• Do you want the image of every dense subset to be dense, or just the one? – Henno Brandsma Dec 28 '13 at 14:35
• of every dense subset – user66906 Dec 28 '13 at 15:04

Not sure about necessary conditions. But if $f$ is continuous and $f[X]$ is dense in $Y$, you already know that $f[A]$ is dense in $Y$, by standard arguments. So bijective is not necessary. Of course $f[X]$ must be dense (as it contains $f[A]$).
• what do you mean by $f[Y]$? – user66906 Dec 28 '13 at 15:13
• Corrected to $f[X]$ – Henno Brandsma Dec 28 '13 at 15:16