Understanding a Math “Meme”

calculusfunctions

I recently saw this "Math Meme":

enter image description here

Loosely speaking, I think I understand the humor – even though some functions may appear to look continuous on paper, they are not mathematically continuous as such.

As an example, I think the "Step Function" (https://en.wikipedia.org/wiki/Step_function) is an example, as I can draw this without lifting my pen/pencil, but is clearly not defined at certain points. Is this a valid example with regards to this math "meme"? If not, can someone please help me come up with one (e.g. perhaps "Absolute Value Function"?)

In general:

  • Is there any particular relevance of the function that they have drawn in this picture (i.e. function plotted in red)?

Is this an example of a discontinuous function?

  • In general, are functions like $$f(x) = x \cdot \sin{\left(\frac{1}{x}\right)}$$ considered discontinuous since at $x=0$ this function is "not defined" (i.e. infinity) – but nonetheless, I can still draw it without lifting my pen?

Thanks!

Note: R Simulation

# Define function
f <- function(x) x*sin(1/x)

# Plot function
plot(f, xlim = c(-5, 5), ylim = c(-5, 5), xlab = "x", ylab = "y")

# Add vertical lines at x = 0 and x = (2*k+1)*pi
abline(v = c(0, pi, -pi, 3*pi, -3*pi), lty = 2)

enter image description here

Warning message:
In sin(1/x) : NaNs produced

Best Answer

A bit of a philosophical answer, or a rant if you like.

I don't like the joke. The gist of it is that the real analysis is filled with cumbersome and confusing definitions of things that are clear and intuitive and obvious to pre-calculus students. Gets you to think - why study real analysis when all it does is muddle waters and make easy stuff hard? Thus the joke comes across as being derogatory towards the real analysis as a subject. (Well... probably the maker of the joke wasn't very good at real analysis in the first place! Or, at least they felt a bit too strongly for those people who were not.)

In fact, if the definition of continuity needed to involve such complicated entities as "I", " can", "draw", " pen" and "lifting", we would have to define those entities first, or give up with using logic in mathematics in the first place! E.g. who says that you can draw the same things as I can; maybe I don't have a pen (does it work with a pencil?) or I am six months old, or I cannot use a pen (if I am too ill), and nobody ever can prove one way or otherwise that $y=\arctan x$ is continuous around $x=10^{100}$. (You can't trick me, this is not $y=\arctan x$, it looks more like $y=\pi/2$ to me!)

So the main point is not in finding example which looks like a function is continuous while rigorously being not, or doesn't look continuous even though it is. The main point is that the pre-calculus definition which includes Earthly creatures (people) and their creations (pens) is not good enough for mathematical use. The "real analysis" definition, warts and all, removes that defficiency at the (small) cost that one needs to study a bit of real analysis to appreciate it.

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