Intuition for Theorems About Continuous Functions

Suppose you're drawing back and forth on a number line (a one-dimensional figure), without ever lifting your pencil. Let's say you start at time $t=0$ (your pencil first touches the line at time $t=0$) and finish at time $t=10$ (your pencil leaves the line at time $t=10$).

After time $t=10$, you look at your beautiful picture. Then the boundedness theorem, intermediate value theorem, and extreme value theorem together state the profound fact that...

... YOU DREW A LINE SEGMENT!

But actually! More formally, let's suppose someone kept track of your pencil the whole time, so your pencil was at position $f(t)$ (on the number line) at time $t$.

Then because you never lifted your pencil, $f(t)$ is a continuous function. Also, since you started at $t=0$ and finished at $t=10$, $f(t)$ is defined for $t$ in the closed interval $[0,10]$.

Then your picture is simply the set of points $f(t)$, for $0\le t\le 10$. Two other ways of writing this are $\{f(t): t\in[0,10]\}$, or more compactly, $f([0,10])$.

Now

(1) The boundedness theorem says your picture is bounded---your picture doesn't go "off to infinity".

More formally, there exist real numbers $A<B$ such that $A<f(t)<B$ for all $t\in[0,10]$.

It's important that $[0,10]$ is a closed interval, because otherwise this picture intuition doesn't apply (even if $f$ is continuous)---you must have started and ended somewhere! (The boundedness theorem is false for open or half-open intervals, as you'll see in the link---but the reason our real-world intuition doesn't work is because you never "could've" drawn any of the counterexamples, even if the universe were infinite, etc. It all boils down to the need for a starting and ending point.)

(2) The intermediate value theorem says your picture has no gaps: if you drew $f(u)$ at time $u$ and $f(v)$ at time $v$, then you drew everything on the number line between $f(u)$ and $f(v)$. In other words, given (1), the intermediate value theorem more or less says that you drew a finite "continuous" interval.

It's important that $f$ is continuous, or in other words that you never lifted your pencil. Otherwise it's easy to come up with counterexamples to the intermediate value theorem, as you can see in the link.

(3) The extreme value theorem says your picture has a rightmost point and a leftmost point. In other words, together with (1) and (2), the extreme value theorem says that you drew a closed interval: a LINE SEGMENT, as claimed earlier!

More formally, $f(t)$ reaches a maximum and minimum on the interval $[0,10]$.

It's important that $[0,10]$ is a closed interval, because otherwise this picture intuition doesn't apply (even if $f$ is continuous and bounded)---you must have started and ended somewhere! (As a silly example, suppose we're considering the function $g(x)=x$ on $(0,10]$, so $g$'s not defined at $x=0$. Well, if we plot $g$ on the number line, then it doesn't have a leftmost point---we're missing $0$. But just as in (1), you never could've drawn this in the first place---if you were able to draw $g(t)=t$ at times $0<t\le 10$, then you'd have to have drawn $0$ at time $t=0$ anyway!)

Intuitively, a continuous function can be thought of a function that can be drawn without raising your pencil off the page. For instance

What is most important to note is that