The solution of ODE is an Integral function, I think it always belongs to the space of continouse functions.

If a function f is a solution to a differential equation in a variable x over a domain D, then f is differentiable with respect to x over D, hence continuous for x in D. Differentiability always implies continuity, but not necessarily the other way around.

If F\in C^{n}, n>=1, then any solutions to u^{\cdot}=F(u) is of class u\in C^{n+1}.

If F is an analytic function so then the solution u.

Fabrizio Morlando Gregory L Wilson @Hao Yin , thank you for your comments