Prove that $f(E)$ is dense in $f(\Bbb R)$

Question

Let $f$ be continuous on $\Bbb R$ and $E$ be a dense subset of $\Bbb R$.

Prove that $f(E)$ is dense in $f(\Bbb R)$

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On the only definition of continuity

For $\forall\ \varepsilon>0$ and $\forall\ x_0\in E$, there exists $\delta>0$ such that $\left|f(x)-f(x_0)\right|<\varepsilon$ for $\forall\ x\in E$ with $\left|x-x_0\right|<\delta$

I think It satisfies condition of dense:

There is $y$ such that $\left|f(x)-f(x_0)\right|<\varepsilon$ for each $y\in f(E)$

But... I'm not sure since it's not clear and and I doubt that only using definition of continuity can prove dense of f(E) so easily.

However, I think it can be solved with only continuous property, because this problem is in chapter of Limit and Continuous.

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This problem is in the textbook which is published only my country (becuase my colleage's professor recently published it in 2021) and there is no solution.

Answer 1

Let $f:\mathbb R\to\mathbb R$ be a continuous function, where $E\subseteq\mathbb R$ is a dense subset.
Since $E$ is dense in $\mathbb R$ we have $\overline E=\mathbb R\implies f(\mathbb R)=f(\overline E)\subseteq\overline{f(E)}$ and this shows that $\forall p\in f(\mathbb R)$, $p\in\overline{f(E)}$ .