Ramanujan's congruences, as extended by Watson and Atkin, show that in the $\ell$-adic metric for $\ell\in\{5,7,11\}$, the partition function is continuous at $\frac{1}{24}$, having a limit $\lim_{n\to_\ell1/24}p(n)=0$. And for $\ell\neq7$, the speed of convergence is linear, so $p$ is Lipschitz continuous at $\frac{1}{24}$.
(I haven't read these proofs, and it's entirely possible that I'm misunderstanding the results. I'm just trying to get a feel for what is known, treating the results as black boxes.)
A natural followup question: Is $p$ also differentiable at $\frac{1}{24}$? (Assume a definition of the derivative appropriate to this context.)
Naively, let $n_1=4, n_2=4+4\cdot 5,n_3=4+4\cdot 5+3\cdot 5^2,\dots$ where $n_k\cdot 24\cong 1 \pmod {5^k}$ and $p(n_k)\to0\;$ $5$-adically. Then it seems $p(n_k)/(n_k-1/24)\cong \pm1\pmod{5}$ depending on if $k\cong 0,1\pmod4$ or $k\cong2,3\pmod4$. Thus the difference quotient does not approach a limit.