Continuous differentiability of transition maps in the definition of a differentiable manifold

Question

I don't understand a requirement of the definition of a differentiable manifold. For an $n$-dimensional manifold $M$ we begin with charts $\varphi:M\to\mathbb{R}^n$, which are homeomorphisms, and require that in an atlas, for any two charts $\varphi$ and $\phi$ that, $\varphi\circ\phi^{-1}$ be continuously differentiable. Why continuously? I get that imposing that condition is the only way to talk about a continuously differentiable map between manifolds, since there needs to be a consistent local notion of continuity of the derivative. However, is there no value to studying manifolds where this desideratum isn't met? Are there things that can be gained from dropping this requirement, for example, some topological manifolds can't be given differential structures, could they be given differential structures without continuously differentiable transition maps?