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”.

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”.