Proving the existence of a non-monotone continuous function defined on [0,1]

Let $({I}_{n}{)}_{n\in \mathbb{N}}$ be the sequence of intervals of [0,1] with rational endpoints, and for every $n\in \mathbb{N}\text{}$ let ${E}_{n}=\{f\in C[0,1]:f\phantom{\rule{mediummathspace}{0ex}}\text{is monotone in}\phantom{\rule{mediummathspace}{0ex}}{I}_{n}\}$. Prove that for every $n\in \mathbb{N}$, ${E}_{n}$ is closed and nowhere dense in $(C[0,1],{d}_{\mathrm{\infty}})$. Deduce that there are continuous functions in the interval [0,1] which aren't monotone in any subinterval.

For a given n, ${E}_{n}$ can be expressed as ${E}_{n}={E}_{n\nearrow}\cup {E}_{n\swarrow}$ where ${E}_{n\nearrow}$ and ${E}_{n\swarrow}$ are the sets of monotonically increasing functions and monotonically decreasing functions in ${E}_{n}$ respectively. I am having problems trying to prove that these two sets are closed. I mean, take $f\in (C[0,1],{d}_{\mathrm{\infty}})$ such that there is $\{{f}_{k}{\}}_{k\in \mathbb{N}}\subset {E}_{n\nearrow}$ with ${f}_{k}\to f$. How can I prove $f\in {E}_{n\nearrow}$?. Suppose I could prove this, then I have to show that ${\overline{{E}_{n}}}^{\circ}={E}_{n}^{\circ}=\mathrm{\varnothing}$. This means that for every $f\in {E}_{n}$ and every $r>0$, there is $g\in B(f,r)$ such that g is not monotone. Again, I am stuck. If I could solve this two points, it's not difficult to check the hypothesis and apply the Baire category theorem to prove the last statement.

Let $({I}_{n}{)}_{n\in \mathbb{N}}$ be the sequence of intervals of [0,1] with rational endpoints, and for every $n\in \mathbb{N}\text{}$ let ${E}_{n}=\{f\in C[0,1]:f\phantom{\rule{mediummathspace}{0ex}}\text{is monotone in}\phantom{\rule{mediummathspace}{0ex}}{I}_{n}\}$. Prove that for every $n\in \mathbb{N}$, ${E}_{n}$ is closed and nowhere dense in $(C[0,1],{d}_{\mathrm{\infty}})$. Deduce that there are continuous functions in the interval [0,1] which aren't monotone in any subinterval.

For a given n, ${E}_{n}$ can be expressed as ${E}_{n}={E}_{n\nearrow}\cup {E}_{n\swarrow}$ where ${E}_{n\nearrow}$ and ${E}_{n\swarrow}$ are the sets of monotonically increasing functions and monotonically decreasing functions in ${E}_{n}$ respectively. I am having problems trying to prove that these two sets are closed. I mean, take $f\in (C[0,1],{d}_{\mathrm{\infty}})$ such that there is $\{{f}_{k}{\}}_{k\in \mathbb{N}}\subset {E}_{n\nearrow}$ with ${f}_{k}\to f$. How can I prove $f\in {E}_{n\nearrow}$?. Suppose I could prove this, then I have to show that ${\overline{{E}_{n}}}^{\circ}={E}_{n}^{\circ}=\mathrm{\varnothing}$. This means that for every $f\in {E}_{n}$ and every $r>0$, there is $g\in B(f,r)$ such that g is not monotone. Again, I am stuck. If I could solve this two points, it's not difficult to check the hypothesis and apply the Baire category theorem to prove the last statement.