Confused about something that has to do with uniform continuity

Question

I am confused about something that has to do with uniform continuity.

The rectangles in the image above represent the regions formed by the following intervals, by the definition of uniform continuity

$$c-\delta<x<c+\delta \Longrightarrow f(c)-\epsilon<f(x)<f(c)+\epsilon$$

The intervals above are illustrated in the graph - makes sense.

However, I am confused about the extremes of the interval in the $y$-axis. Namely, $f(c)-\epsilon \text{ and } f(c)+\epsilon$.

They do not seem like actual outputs that are mapped to the line of the function. Consider the first rectangle. If you start at the top edge of the rectangle ($f(c)+\epsilon$) and try to move horizontally to the point that it is mapped to on the line, there isn't one. There is no point on the line for ($x+\delta$, $f(c)+\epsilon$).

This is what I am confused about. How does this work? I know there is something I am either overthinking or just confused about. I would appreciate an explanation, elaboration, or clarification.

Thank you.

Image from WIKIMEDIA COMMONS by Claudia4 " Gleichmäßig stetige Funktion, visualisiert durch epsilon-delta-Umgebungen gleicher Größe" 22 Sept. 2016

Answer 1

Those are the boxes in which the values of the function must lie if that $\epsilon$ will work for all input values. In the picture the graph of the function does in fact remain in those boxes.

The "uniform" here is reflected by the fact that the boxes all have the same width. Just plain continuity would call for a box at each point whose width (the value of $\delta$) might vary.

Answer 2

The graph of the function, when restricted on $(x-\delta,x+\delta)$, is a subset of one of the gray rectangles in your picture. In each rectangle, only the green part is part of the graph of the function.

The inequality $f(c)-\epsilon<f(x)< f(c)+\epsilon$ simply says that $f(x)$ falls in the interval $(f(c)-\epsilon, f(c)+\epsilon)$. It is not necessarily true that $f(x)$ achieves the value, say, $f(c)+\epsilon$.

Consider the example when $f$ is a constant function $f(x)=1$. For any $x$, in any rectangle, $f(x)$ takes only one value in the interval $(f(c)-\epsilon, f(c)+\epsilon)$.