When we say $f \in C^1$, we mean that $f$ is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So why in most of the textbooks they always mention them two?
So these are all equivalent:
- $f \in C^1$
- $f$ is continuously differentiable
- $f'$ exists