I know . If $f:[a,a+h]\rightarrow \mathbb{R}$ continuous and differentiable on $(a,a+h)$ exists $t$ , $0<t<1$ s.t. $f(a+h)=f(a)+ f'(a+th)h$
But how could become $f(x)\in {C[0,a]}$, $D^{\alpha}f(x)\in C[0,a]$ for $0<\alpha\leq 1$ Then we have $f(x)=f(0^{+}) + \dfrac{1}{\Gamma (\alpha)}(D^{\alpha}f)(\xi).x^{\alpha}$ with $0\leq \xi \leq x$ $\forall x\in (0,a]$
I would like to know how I can prove this.
Where
$D^{\alpha}f(x)=\dfrac{1}{\Gamma (\alpha)}\int_{0}^{x} \! (x-t)^{\alpha-1}f(t) \, dt = 8$. is Riemann Liouville integral.
Thanks!