OK, playtime is over. This one, like the still unresolved WitCH 8, will take some work. It comes from Cambridge’s ** Mathematical Methods 3 & 4 (2019).** It is the introduction to “When is a function differentiable?”, the final section of the chapter “Differentiation”.

## Update (12/08/19)

We wrote about this nonsense seven long years ago, and we’ll presumably be writing about it seven years from now. Nonetheless, here we go.

The first thing to say is that the text is wrong. To the extent that there is a discernible method, that method is fundamentally invalid. Indeed, this is just about the first nonsense whacked out of first year uni students.

The second thing to say is that the text is worse than wrong. The discussion is clouded in gratuitous mystery, with the long-delayed discussion of “differentiability” presented as some deep concept, rather than simply as a grammatical form. If a function has a *derivative* then it is *differentiable*. That’s it.

Now to the details.

The text’s “first principles” definition of differentiability is correct and then, immediately, things go off the rails. Why is the function f(x) = |x| (which is written in idiotic Methods style) not differentiable at 0? The wording is muddy, but example 46 makes clear the argument: f’(x) = -1 for x < 0 and f’(x) = 1 for x > 0, and these derivatives don’t match. This argument is unjustified, fundamentally distinct from first principles, and it can easily lead to error. (Amusingly, the text’s earlier, “informal” discussion of f(x) = |x| is exactly what is required.)

The limit definition of the derivative f’(a) requires looking precisely *at *a, at the gradient [f(a+h) – f(a)]/h as h → 0. Instead, the text, with varying degrees of explicitness and correctness, considers the limit of f’(x) *near *a, as x → a. This second limit is fundamentally, conceptually different and it is not guaranteed to be equal.

The standard example to illustrate the issue is the function f(x) = x^{2}sin(1/x) (for x≠ 0 and with f(0) = 0). It is easy to to check that f’(x) oscillates wildly near 0, and thus f’(x) has no limit as x → 0. Nonetheless, a first principles argument shows that f’(0) = 0.

It is true that if a function f is continuous at a, and if f’(x) has a limit L as x → a, then also f’(a) = L. With some work, this non-obvious truth (requiring the mean value theorem) can be used to clarify and to repair the text’s argument. But this does not negate the conceptual distinction between the required first principles limit and the text’s invalid replacement.

Now, to the examples.

Example 45 is just wrong, even on the text’s own ridiculous terms. If a function has a nice polynomial definition for x ≥ 0, it does *not* follow that one gets f’(0) for free. One cannot possibly know whether f’(x) exists without considering x on both sides of 0. As such, the “In particular” of example 46 is complete nonsense. Further, there is the *sotto voce* claim but no argument that (and no illustrative graph indicating) the function f is continuous; this is required for any argument along the text’s lines.

Example 46 is wrong in the fundamental wrong-limit manner described above. it is also unexplained why the magical method to obtain f’(0) in example 45 does not also work for example 46.

Example 47 has a “solution” that is wrong, once again for the wrong-limit reason, but an “explanation” that is correct. As discussed with Damo in the comments, this “vertical tangent” example would probably be better placed in a later section, but it is the best of a very bad lot.

And that’s it. We’ll be back in another seven years or so.

Just a question (genuine, I promise!) Mathematica gives “ComplexInfinity” as the limit for this example. Is the word “complex” just dressing, or does it convey additional meaning? (Anyone is free to assist me here)

Example 47? Roots can always be considered as multiple-valued complex things, but isn’t there a way to tell Mathematica to not be dumb?

Of course, the specification Reals can be used in any solve command (amongst other things) but I was asking whether “complex” infinity was any different to the Aleph-Null infinity that comes to mind when using the word “infinity” more loosely.

Ah, I see. Yes, complex infinity is a thing, and it is different. In the reals you can go to +∞ or -∞. In complex numbers you can think of going out along any radial direction, or weird spirals or whatever, but it turns out to be useful to think of just one point ∞ at the end.

In Example 46 they say:

“f ‘(0) is not defined as the limits from the left and the right are not equal.”

I think that it is fair to assume that they mean that the expression for f'(x) when x>0 is not equal to the expression for f'(x) when x0. And I think that it is fair to assume that they assume that this is equivalent to the very first line of this WitCH, where they define what it means for a function to be differentiable. But it’s not.

Thanks, Damo. You’ve nailed the central problem, though there’s lots more there. The whole discussion is inconsistent and strange, and it is worth noting the “informal treatment” in section 9D to which the text refers (Example 19). But there is no question that the “limits” in Example 46 refers to limits of derivatives. As you point out, that simply does not give an equivalence for differentiability.

I do not feel confident about Example 47, but I’m going to give it a go. To begin with, to show the graph of the derivative of f(x) and not the graph of f(x) is pretty crap. There should at least be some discussion of the fact that at x=0 there is a vertical tangent line. Also, returning to their initial definition of differentiability, I’m not sure why they didn’t use limits here. I mean, they kinda do, but they don’t. And actually, that limit is defined as being equal to infinty. So where does that leave us? You can either extend the reals to include +infinity and -infinity, in which case the function is differentiable, or you need to return to the original definition and say that it is only differentiable if the limit exists and is finite. But to gloss over it and not even relate it back to the original function? Not ideal.

Not ideal? Damo, you’re a master of understatement. It’s a godawful mess. To take it bit by bit:

1) Yes, of course they should show the graph of f as well as that of f’. The solutions to Examples 45 and 46 are similarly flawed.

2) Yes, it is a missed opportunity to not bring in the notion of vertical tangent here. True, that concept is only introduced (poorly) in the next chapter, but this simply demonstrates that Example 67 is misplaced. What does it do in this section other than confuse?

3) It is standard to discuss “infinite” limits without extending the reals (though it’s better to bite the bullet and extend). But yes, flippantly throwing in an infinite limit here is close to meaningless.

4) Yes, they correctly look at the correct limits (give or take infinity), but only as an afterthought, as an “explanation” of their (invalid) “solution”.

I am going to try and be a little generous here – while their approach to this is a shambles, this specific example and the explanation of why we can or can’t differentiate when x=0 is done universally badly. Even when reference is made to a vertical tangent, the argument that is often made for it not being differentiable is that the slope of this tangent is undefined or simply that f'(x) is not defined when x=0. They have at least made some attempt to refer back to the original definition, although, as you point out, it is done as an afterthought and with no

respect for the significance of the result.

On another matter, the casual use of smoothness in the original notes and the questions leaves me anxious.

On another matter, this sudden influx of WitCHs has left me feeling giddy, like it’s Christmas. At some point, I’m going to have to begin engaging with my family again, but for now, this is too much fun. Thank you.

Damo, you set a good example, and I can also be generous here. But I choose otherwise.

I have sympathy for the writers. It is very tricky to write correctly and clearly on informal limits, and infinite and non-existent limits are trickier still. If the “explanation” for Example 47 had appeared in isolation, I’d be fine with it. But, it’s *not* in isolation, and it’s not an “explanation”, and it’s not an “also”.