Showing a set is nowhere dense in $C([0,1])$

Solution 1:

I assume you mean $C[0,1]$ with sup-norm.

Hint: Try to show that for each $n$ and arbitrary $\varepsilon>0$ there exists $g\notin E_n$ such that $\|g\|_\infty<\varepsilon$.

Try to show that $f\in E_n$ and $g\notin E_{2n}$ implies $f+g\notin E_n$.

Using these two facts you should be able that if $f\in E_n$ then in any ball $B(f,r)$ there is a function which does not belong to $E_n$.

Solution 2:

Let $g(x)=\lvert x\rvert$ in $[-1,1]$, and extend $g$ to be periodic in $\mathbb R$, with period $2$. Set $$ g_{k,\ell}(x)=\frac{1}{k} g(\ell x). $$ It is not hard to see that $$ g_{k,\ell}\in E_n \quad\text{iff}\quad n\ge \frac{\ell}{k}. $$ Fix now $n\in\mathbb N$. We shall show that $E_n$, which is a closed subset of $C[0,1]$, has empty interior.

Let $f\in E_n$ and $\varepsilon>0$. We shall show that $B(f,\varepsilon)\not\subset E_n$. Let $k,\ell\in\mathbb N$, so that $$ \frac{1}{\ell}<\varepsilon\quad\text{and}\quad \frac{k}{\ell}>2n. $$ Then $$ f+g_{k,\ell}\in B(f,\varepsilon) \quad\text{and}\quad f+g_{k,\ell}\not\in E_n. $$