Here's a continuity result that I believe to be true, but I don't know if my assumptions are minimal (i.e. does it still go through with just assuming continuity or something slightly weaker conditions like Lipschitz continuity?), let alone a valid proof. I'm sure someone's proved a continuity result like this before, but I couldn't find any sources that deal specifically with a lot of basic continuity results. Does anyone know any sources (or an ensemble of sources) that deal with a bunch of useful continuity results? Here's the basic property that I believe to be true:
Let $f:[a,b]\mapsto\mathbb{R}$.
If $f$ is a continuously differentiable function on $[a,b]$ and $0\leq\alpha\leq 1$, then $D_{a+}^{\alpha}(f(t))$ is continuous on $(a, b]$ (here I'm using the Riemann-Liouville definition).
Proof:
$D_{a+}^{\alpha}(f(t))=\frac{d}{dt}(\frac{1}{\Gamma(1-\alpha)}\int_{a}^{t}(t-\tau)^{-\alpha}f(\tau)d\tau)$
So all that's left to prove is that the expression in the inside of the derivative operator is continuously differentiable on (a, b]. So since we assumed continuous differentiability of f, we can integrate by parts, obtaining
$ \frac{1}{1-\alpha}(t-\tau)^{1-\alpha}f(\tau)|_{a}^{t} - \frac{1}{1-\alpha}\int_{a}^{t}(t-\tau)^{\alpha-1}f'(\tau)d\tau $
The term with the integral is continuously differentiable because the integrand is continuous. So the term without the integral becomes
$ (t-a)^{1-\alpha}f(a) $
whose derivative is $(1-\alpha)(t-a)^{-\alpha}f(a)$, which is continuous as long as $t>a$, so is continuous on $(a,b]$, completing the proof.