Complicated problem of Real analysis and set theory from NBHM 2006

Let $f$ be a real-valued function on $\Bbb{R}$. Consider the functions

$$w_j(x) = \sup \left\{ \left|f(u)-f(v)\right| : u,v \in \left[x-\frac{1}{j},x+\frac{1}{j}\right] \right\}$$

where $j$ is a positive integer and $x \in\Bbb{R}$. Define next

$$A_j,_n = \left\{x \in \Bbb{R}: w_j(x)\lt \frac{1}{n}\right\}, n=1,2,3,...$$

and

$$A_n = \underset{j=1}{\overset{\infty}\cup}A_{j,n}, n=1,2,3,...$$

Now let $C= \left\{x \in \Bbb{R} : f \text{ is continuous at } x \right\}$.

Express $C$ in terms of the sets $A_n$.

Answer given in solution set as $C = \underset{n=1}{\overset{\infty}\cap}A_n$

So this question was asked in 2006 NBHM PhD scholarship exam (India). I have tried to understand it but failed;

then I tried using trivial functions like constant function and Identity function ( which are continuous on $\Bbb{R}$ ).

When I took $f$ equal to the constant function, I got $w_j(x) = \{ 0 \}$ for each $j$

and then $A_{j,n} = \Bbb{R}$, and hence $A_n=\Bbb{R}$ for each $n$.

And hence $C$ (here$\Bbb{R}$) can be written as an intersection of $A_n$'s.

When I tried $f$ as the Identity function, calculations became more complicated and eventually, I gave up.

I know that this problem should not be solved by using examples,

I have to find a solution which will work for every real-valued function (Continuous or Discontinuous).

But I'm unable to do so. Please help.

Answer

First lets write down a definition of continuity. A function $f$ is continuous at $x$ iff for every $n \in \mathbb{N}$ there exists a $j \in \mathbb{N}$ such that $\sup \{ |f(u) - f(v)| : u,v \in [x- \frac{1}{j}, x + \frac{1}{j}] \} < \frac{1}{n}$. (This is worded a little differently to the usual $\varepsilon$-$\delta$ definition but it's fairly easy to see it's equivalent.)

Now we simply reword this in terms of the sets in your question. First note that for fixed $n \in \mathbb{N}$ there exists a suitable $j$ iff $x \in A_{j,n}$ for some $j \in \mathbb{N}$ or equivalently iff $x \in A_n$. So $f$ is continuous at $x$ iff there is a suitable $j$ for each choice of $n$ which is exactly when $x \in \bigcap_{n \in \mathbb{N}} A_n$ which is the desired result.