Family of function with fractional derivatives

I would like a family of functions $f_a(x)$ so that $f_a$ is $a\in\mathbb{R}$ fractionally differentiable but not $a+\epsilon$ fractionally differentiable.

Does anyone know such functions which are easily described?

Thanks in advance.

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1 Answer

x^a is fractionally differentiable at $x=0$ with $b$-fractional derivative proportional to $x^{a-b}$ for $b \leq a$. However, if $b > a$, then the fractional derivative blows up at the origin.

If you want a collection of functions which have $a$-fractional derivatives but such that the $b$-fractional derivatives for $b>a$ simply do not exist, consider the $a$-fractional anti-derivatives of $|x|$.

These are easy to find, because $|x| = x$ for $x>0$, and $|x| = -x$ for $x \leq 0$, and so you can just use the formula for the $k$th anti-derivative of $x$.

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Do you also have an example which doesn't blow up? – alext87 Oct 31 '10 at 8:42
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