Show that $C[a,b]$ is separable with supremum norm using piecewise linear functions

Question

Show that $C[a,b]$ is separable with supremum norm by showing countable and dense subset of piecewise linear functions.

Since I cannot use Stone-Weierstrass, I have been struggling with construction and approximation error.

For any given function $f \in C[a,b]$ and for fixed $\epsilon>0$ I have to construct piecewise linear function $p(x)$ s.t. $\sup_{x\in[a,b]}|f(x)-p(x)| < \epsilon$.

Now I have two important questions. How do we construct such piecewise linear function? And moreover, how do we find that the error is lower than epsilon?

Is it somehow connected with the fact that $f$ is also uniformly continuous?

Answer 1

It has a lot to do with uniform continuity.

Take $\varepsilon>0$. There is a $\delta>0$ such that$$\bigl(\forall x,y\in[0,1]\bigr):|x-y|<\delta\implies\bigl|f(x)-f(y)\bigr|<\frac\varepsilon3.$$Pick rational numbers $a_0=0,a_1,a_2,\ldots,a_n=1$ with $a_0<a_1<\cdots<a_n$ and that$$\bigl(\forall i\in\{1,2,\ldots,n\}\bigr):a_u-a_{i-1}<\delta.$$For each $i\in\{1,2,\ldots,n\}$, let $b_i\in\mathbb Q$ be such that $\bigl|b_i-f(a_i)\bigr|<\frac\varepsilon3$. Let $g$ be the piecewise linear function which is linear on each interval $[a_{i-1},a_i]$ and such that $g(a_i)=b_i$ for each $i\in\{0,1,\ldots,n\}$. Then $(\forall x\in[0,1]):\bigl|f(x)-g(x)\bigr|<\varepsilon$. FInally, the set of all such piecewise linear functions $g$ is countable.

Answer 2

I'll let you do the details, but here is the idea: since $f$ is uniformly continuous, can partition $[a,b]$ in small intervals (using $\delta$) so that the value of $f$ doesn't change more than $\varepsilon$ inside any of those intervals. At this stage, you have found a step function that is at less that $\varepsilon$ from $f$. Now you want to "connect the steps" to make the step function continuous.

Answer 3

FYI, the lattice version of the Stone-Weierstrass theorem will help you here. Note that piecewise linear functions form a lattice under the usual pointwise ordering, in the sense the minimum and maximum of two piecewise functions are piecewise linear. It's also very easy to construct a (piecewise) linear function that connects any two points.