Let $f \in C^k \mathbb R, f: \mathbb R \to \mathbb R$, where $f$ may or may not belong to a higher differentiability class. Then the derivatives $f^1, f^2, ..., f^k$ exist, and $f^k$ is continuous. Each $f^i$, which I understand is such that $f^i \in C^{k-i}\mathbb R$, may or may not belong to a higher differentiability class. Do we have the following?
$$f \in C^{\infty} \iff f^1 \in C^{\infty} \iff ... \iff f^k \in C^{\infty}$$
I'm wondering because I haven't seen anything like that on wikipedia or stackexchange. Maybe it's so obvious that no one mentions it, or it's wrong.
If $f^{(k)}$ (that's how I have learned to denote the $k$th derivative) is not smooth, then that must be because for some integer $n$ it doesn't have a continuous $n$th derivative. Now try to differentiate $f$ a total of $(k+n)$ times to see that $f$ isn't smooth either.