Mark Ryan, in his book, Calculus for Dummies, defines the following function:

$$ g(x)= \begin{cases} x\sin\frac{1}{x}, & \text{if $x \ne 0$} \\ 0, & \text{if $x=0$ } \end{cases}$$

He then uses the Squeeze Theorem to show that $$\lim_{x\to 0} g(x)=0$$

Here's a reproduction of his figure, where $f(x)=|x|$ and $h(x)=-|x|$.

Mark Ryan graph

What happens as $x \to 0$?

Here are my thoughts:

Clearly, the value of the function approaches $0$. However, Ryan, having extended the domain to include $0$, mischievously asks what the function is doing at $0$. Is it increasing or decreasing? He writes, "Stuff like this really messes with your mind."

Now, I suspect there is no answer to the question as he's asked it. Since the length of the line approaches infinity as $x$ approaches $0$, we may as well ask what happens at infinity.

However, I find that response dissatisfying. It reminds me of when I was in school, and was told that $a/0$ was undefined. I felt it $should$ be infinity. So, when I explained to a high school student recently that while he couldn't divide $3$ by $0$, I also showed him a graph of $y=3/x$. I explained that what we could do was talk about what happens to the value of $y$ as the value of $x$ approaches the $y$-axis from the left and right sides.

In other words, while we can't ask about the value of $3/0$, there is something interesting we can ask that is analogous to that question. We can't scratch the exact itch, but we can scratch nearby.

So, my question is this:

a) Is it true that Ryan's question has no answer, even though $f(x)$ is defined?

b) If the question does have no answer, is there a similar, perhaps more interesting question we can ask about the function at $x=0$?

  • $\begingroup$ This is what known as Topologist's sine curve. $\endgroup$ – Kushal Bhuyan Sep 19 '15 at 5:34
  • $\begingroup$ @KushalBhuyan - My apologies. The function in the body of the question is what I was talking about. I've edited my question. Having said that, the topologist's sine curve poses the same question. Thanks for drawing it to my attention. $\endgroup$ – Adam Hrankowski Sep 19 '15 at 15:47
  • $\begingroup$ @KushalBhuyan The topologists' sine curve is parametrised by $(x, \sin(1/x))$. Not quite the same thing. $\endgroup$ – Patrick Stevens Sep 19 '15 at 15:50

Ryan's question has an answer: $f$ is neither increasing nor decreasing at $0$. Inside any interval about $0$, there are values $x$ to the left of $0$ with $f(x) > f(0)$ and other values where $f(x) < f(0)$, and similarly to the right of $0$. Thus neither description applies to $f(x)$ at $0$.


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