Once Continuously Differentiable?

Question

I'm hoping someone can clarify this because I can't seem to find a solid answer.

I know functions can be continuously differentiable, but I just read in a textbook "this applies to once continuously differentiable..." but they don't give an example and google seems unhelpful.

Am I right in assuming that once continuously differentiable is equivalent to continuously differentiable?

If not can you please provide an example that clearly demonstrates the difference.

Answer 1

Another simple example

$$x\mapsto \cases{0 & if $x < 0$\\x^2 & otherwise}$$

Answer 2

Once continuously differentiable is indeed equivalent to continuously differentiable, but it emphasis the point that the function may not be more than once continuously differentiable. For example : $$x\mapsto \cases{0 & if $x = 0$\\x^3\sin\left(\frac{1}{x}\right) & otherwise} $$ is exactly one time continuously differentiable.