Does this symbol $C^1([0,1],\mathbb{R})$ means that if $f$ in $C^1([0,1],\mathbb{R})$ then it is continuous on $[0,1]$ and its first derivative $f'$ is also continuous on $[0,1]$?
I have the function $f(x) = x^{1/3}$ as an example that troubles my mind.
Any explanation will be helpful!
It means that $f$ is differentiable in $[0,1]$ and its derivative is continuous. Note that this automatically implies that $f$ itself is continuous, so you don't have to mention that in the definition of $C^1$.
In general, $C^k([0,1],\mathbb{R})$ is the space of functions from $[0,1]$ to $\mathbb{R}$ that are differentiable $k$ times and their $k$th derivative is continuous.