What are the extremum points on this graph?

Question

The graph in question is here.

Assuming the inspected interval is $[0, 4]$:

1- The point $(0, 4)$ does not qualify to be an absolute maximum, but what about say $(0.000\cdots001, 3.9999\cdots)$? Why can't we say that the interval $[0, 4]$ has an absolute maximum even though there are relatively bigger values between $x = 0$ and $x = \frac13$?

2- Can we say that the point $(0, 1)$ is an absolute minimum for this interval?

3- The point $(1, 3)$ is a local maximum, right?

4- Lastly, what about the point $(4, 3)$? Can we say it's a local maximum too?

My thoughts are: Questions 2 and 3 can most likely be answered with "yes". I've seen many conflicting answers regarding question 4, so I'm not sure anymore...

And question 1 is the one on which I couldn't find any relatable answers. The endpoint x = 0 where the function would normally have a maximum is not continuous. Therefore, appointing the "next closest" maximum value $(3.999\cdots)$ seemed logical enough, but apparently it doesn't work that way. But why is that? My guess would be that we can't define it as an exact point, therefore we dismiss it..? I would love some clarification on this.

Answer 1

1. There is no such point. For any point in the interval that you claim to be an absolute maximum, I can produce a point at which the graph is higher.
2. Yes.
3. Yes.
4. Yes.

Therefore, appointing the "next closest" maximum value $(3.999⋯)$ seemed logical enough, but apparently it doesn't work that way. But why is that?

There is no "next closest" point: for any non-zero $x$, there is some point between $0$ and $x$ (indeed, there are uncountably infinitely many such points). $3.999\ldots$ is exactly equal to $4$.