Why do authors make a point of $C^1$ functions being continuous?

Question

I've just got a little question on why authors specify things the way they do. Is their some subtlety I'm missing or are they just being pedantic?

I've encountered the function spaces $C^k[a,b]$ a few times this year and usually the author will make a point that the functions are continuous and have continuous first derivatives, continuous second derivatives, and so on up to $k$. Why bother specifying it like this? A differentiable function is necessarily continuous so couldn't we just state $C^k[a,b]$ as the space of real/ complex -valued functions with continuous $k$th derivatives? Then the functions themselves and their less-than-$k$th derivatives would have to be continuous as well.

Answer 1

You are right. If a function has a k-th derivative it must have a 1st, 2nd,...,(k-1)-th derivative and these have to be continuous. So it is not necessary to mention this. Wolfram Mathworld starts its article about $C^k$ functions with

A function with k continuous derivatives is called a $C^k$ function. ...

Of course one also could say

A function with a continuous k-th derivative is called $C^k$ function. ...

Answer 2

As Dave Renfro commented, this may be useful for pedagogical reasons, even if it's logically unnecessary. One difficulty that people often have is putting too much trust in formulas. Of course if $f'$ is to exist, $f$ must be continuous, but a formula for $f'$ might sometimes exist and be continuous without $f$ being continuous. If you don't first check for discontinuities of $f$, you might miss them. As a simple example, consider $$ f(x) = \arctan(\tan(x))$$ The naive student, asked to check if $f$ is $C^1$, might start by computing with the Chain Rule $$ f'(x) = 1 $$ see no sign of discontinuity there, and conclude that $f$ is $C^1$.

Of course, it's easy for a somewhat less naive student to see the error in this case, but more complicated examples can arise that can even catch experts off guard. For example, a "closed-form" antiderivative of a meromorphic function will typically have branch cuts, even though the poles are not real; whether the branch cuts intersect the real axis, and if so where, is often not obvious. This often arises with antiderivatives produced by Computer Algebra systems.