Problem 1.6.1 from Berkeley Problems in Math

Question

Problem 1.6.1 (Fa84) Prove or supply a counterexample: If f is a nondecreasing real valued function on $[0, I]$, then there is a sequence of continuous functions on $[0,1]$, ${f_n}$, such that for each $x \in [0,1]$, $\lim_{n\rightarrow \infty} f_n(x) = f(z)$.

Start of Solution

Solution to 1.6.1: Let B be the set of function that are the pointwise limit of continuous functions defined on $[0,1]$. The characteristic functions of intervals, XI, are in B. Notice also that as f is monotone, the inverse image of an interval is an interval, and that linear combinations of elements of B arc in B. Without loss of generality, assume $f(0) = 0$ and $f(1) = 1$.

Even though the solution is in front of my eyes, I am having trouble deciphering it:

1. How do you know to look at the set of pointwise limits of continuous functions? Can anyone provide an intuitive understanding of what this limit set looks like?
2. Characteristic functions - wouldn't these be just $0$ or $1$ - basically the same for each interval XI? So, how does this help?
3. Without loss of generality, assume $f(0) = 0$ and $f(1) = 1$. Sounds like a loss of generality to me.

Thanks AV

Answer 1

Hint: let $x_0,x_1,\dots$ be the discontinuity points. Considet the function $l(x)=\sum\limits_{j=1}^\infty s_n(x)(\lim\limits_{z\to x_n^+}f(z)-\lim\limits_{z\to x_n^-}f(z))$. where $s_n(x)= 1$ if $x\geq x_n$ and $0$ otherwise.

Prove that $f-l$ is a continuous function.

So now all that you have to do is show that there is a sequence $l_1,l_2,\dots$ of continuous functions that converge to $l$ ( because then $f-l+l_1,f-l+l_2,\dots$ converges to $f$).

But is is easy to prove that there is a sequence of continuous functions that converges to $l$.

remember that we can approximate $s_n(x)$ with the function $t_{n,k}(x)=\min(1,(\frac{x}{a})^k)$

So the function $l_k=\sum\limits_{i=1}^k t_i,k(x)(\lim\limits_{z\to x_n^+}f(z)-\lim\limits_{z\to x_n^-}f(z))$ does the trick.