Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to find examples of functions that are in $C^2(\mathbb{R})$ but not in $C^3(\mathbb{R})$. I am also wondering about $C^2[0,1]$ and $C^3[0,1]$.

If I am not mistaken about the definitions, a function is said to be in $C^p(A)$ if it is a real-valued function with the set $A$ as its domain and its $p$-th derivative being defined on all of $A$ and continuous.


share|cite|improve this question
The definition I think that you mean is that its $p$th derivative is a continuous function. – alex.jordan Nov 16 '12 at 7:57
Thanks Alex. I will re-edit the definition. I was not really sure about it! – Learner Nov 16 '12 at 7:59
I think your original definition is fine if you replace the typo "differential" with "differentiable". – alex.jordan Nov 16 '12 at 8:00
Alex, do you think the current definition is ok now? – Learner Nov 16 '12 at 8:01
I'd say so. You could emphasize that the $p$th derivative is also defined on all of $A$. And another thing: it's all just about convention, but I would have it that by "differentiable on $[0,1]$" you are referring to a topological definition of the derivative that uses open neighborhoods, whereby differentiable at $0$ or at $1$ means what would normally be a one-sided derivative. – alex.jordan Nov 16 '12 at 8:06
up vote 2 down vote accepted

Take some function $f \in C^0(\mathbb R) \setminus C^1(\mathbb R)$, for example $f\colon x \mapsto \left|x-\frac 12\right|$. Taking the antiderivative twice, gives you $$ F \colon x \mapsto \int_0^x \int_0^\xi \left|t - \frac 12\right| \, dt\,d\xi $$ which is in $C^3(\mathbb R) \setminus C^2(\mathbb R)$ (no, you don't want $C^2(\mathbb R)$-functions to be bounded). Restricting $F$ to $[0,1]$ gives you an example there.

share|cite|improve this answer

Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by $$ f(x):=x^2|x|,\text{ for all }x\in\mathbb{R}. $$ By translating the function suitably at $\frac{1}{2}$, one can answer the next question as well.

share|cite|improve this answer

Take a function that is piecewise continuous but not continuous and antidifferentiate it the proper number of times.

For example, let $f(x)=\begin{cases}1&x>0\\-1&x<0\end{cases}$. A first antiderviative is $|x|$. A second antiderivative is $x|x|/2$, and a third antiderivative is $x^2|x|/6$. This last function meets your conditions.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.