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In this paper, we use local fraction derivative to show the H\"oldercontinuity of the solution to the following nonlinear time-fractional slow andfast 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 thecondition 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, wehave removed this extra condition, which in particular includes all cases for$\beta\in(0,2)$.

H\"older regularity of the nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$