Let us start with the definitions that $\frac{d}{dx}\theta(x)=\delta(x)$ where $\theta(x)$ and $\delta(x)$ are Heaviside and delta functions.
Now, with the definition: $^c D^\alpha f(t)=\frac{1}{\Gamma(n-\alpha)}\int_0^t\frac{\frac{d^n}{dx^n}f(x)}{(t-x)^{\alpha-n+1}}\mathrm d x$ I want to understand what is the half-derivative of the $\theta(x)$ such that applying another half derivative on the result gives the delta function.
For a simple function like say $f(t)=sin(x)$ one can look at the Taylor expansion and for each power of x, the formula gives the half derivative to be $$^c D^\frac12 x^p=\frac{p x^{p-\frac{1}{2}} \Gamma (p)}{\Gamma \left(p+\frac{1}{2}\right)}$$
such that application of another half derivative gives the same result as the full derivative i.e. $px^{p-1}$. However, for the Heaviside theta function, the first application gives $$^c D^\frac12 \theta(x)=\frac{2 \theta (x)-1}{\sqrt{\pi } \sqrt{x}}$$
and now I am usure how to show that the second application gives the Delta function.
Too long for a comment:
The derivative of a distribution $\theta$ is defined as $$ ( \partial_x \theta, \varphi):= -(\theta, \partial_x \varphi) $$ For any function $\varphi \in C^\infty_c$ (or if the distribution has compact support $C^\infty$)
As such the most natural definition for the fractional derivative of a distribution will be
$$ ( ^cD^{\frac{1}{2}} \theta, \varphi):=i(\theta, ^cD^{\frac{1}{2}} \varphi) $$ for any smooth (with compact support) $\varphi$.
It is immediate to see that $\left( ^cD^{\frac{1}{2}}\right)^2\theta=\partial_x\theta$ for any distribution $\theta$
I would be surprised if the fractional derivative of the heaviside function is a function and not a distribution as it would be
$$ (^cD^{\frac{1}{2}} \theta, \varphi)= \frac{i}{\Gamma(n-\frac{1}{2})} \int_{\mathbb{R}} \theta(t) \int_0^t\dfrac{\partial_x^n \varphi(x)}{(t-x)^{a-n-1}} dx dt= $$
$$=\frac{i}{\Gamma(n-\frac{1}{2})}\int_0^\infty \int_{0}^t \dfrac{\partial_x^n \varphi(x)}{(t-x)^{a-n-1}} dx dt $$ That does not look as it can be written as a function