# Once Continuously Differentiable?

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.

• Yes, what this is emphasising is that one needs not assume the function is twice continuously differentiable, that is there is no need to assume its derivative also has a continuous derivative. – Angina Seng Mar 6 '18 at 10:56

$$x\mapsto \cases{0 & if x < 0\\x^2 & otherwise}$$
• Yes this is simpler to see and assuming $x^n$ we can find examples for $n-1$ times continuously differentiable. You are welcome! Bye – user Mar 6 '18 at 11:28
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.
• @KennyB : if you compute the derivative of it over $\mathbb{R}\setminus \{0\}$ you will have $2x\sin(1/x)-x \cos(1/x)$ which you can show it is the derivative over $\mathbb{R}$ by taking $0$ at $x=0$, but if you derivate again, you will have terms like $\sin(1/x)$ alone and it can't be continuous at $0$. – Netchaiev Mar 6 '18 at 12:43