"Every elementary function is differentiable."

Question

Edwards and Larson (Calculus, 2018) claim that:

You can differentiate any elementary function.

It seems though that this claim is false.

How then can we modify the above claim so that it becomes true?

Perhaps for example it is true that every elementary function is differentiable except on a set of isolated points?

(ProofWiki definitions of elementary function and differentiability.)

Answer 1

If you interpret "you can differentiate any elementary function" as "every elementary function is differentiable on $\mathbb R$", then the statement is false.

However, the book actually says something else:

With the differentiation rules introduced so far in the text, you can differentiate any elementary function. For convenience, these differentiation rules are summarized below. [derivatives table is given]

which is very likely to be true (I didn't read the book, so it's an assumption), as in:

No techniques beyond those already introduced are necessary to arrive at the list of differentiation rules, applicable to all elementary functions where they are differentiable, that we are about to list below.

The important thing when reading mathematics is to be aware that some statements will be vague and not really correct when taken too literally. However, they are not required to be, unless they are specifically tagged to be definitions or theorems. Just read with grain of salt and you'll be fine.

Answer 2

An elementary function is in the first place a function term composed using certain rules: you may use a single variable $x$, complex constants, $+$, $-$, $*$, $:$, $\exp$, $\log$, $\circ$, etc., but not "cases", as in the definition of ${\rm abs}:\>{\mathbb R}\to{\mathbb R}$. When such a function term defines an actual real or complex function on a reasonable subset of ${\mathbb R}$ this function is called elementary as well. Note that the function $x\mapsto {1\over x}$ is a bona fide elementary function even though it is undefined at $x=0$.

When it is said that "all elementary functions are differentiable" the meaning is that the derivative of a legal elementary function term, computed according to the rules, is again a legal elementary function term.