  In this paper, we use local fraction derivative to show the H\"older
continuity of the solution to the following nonlinear time-fractional slow and
fast diffusion equation:
  $$\left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) =
I_t^\gamma\left[\sigma\left(u(t,x)\right)\dot{W}(t,x)\right],\quad t>0,\:
x\in\mathbb{R}^d,$$ where $\dot{W}$ is the space-time white noise,
$\alpha\in(0,2]$, $\beta\in(0,2)$, $\gamma\ge 0$ and $\nu>0$, under the
condition that $2(\beta+\gamma)-1-d\beta/\alpha>0$. The case when
$\beta+\gamma\le 1$ has been obtained in \cite{ChHuNu19}. In this paper, we
have removed this extra condition, which in particular includes all cases for
$\beta\in(0,2)$.
