= Solution
The continuous representative of $u$ makes $x\mapsto2xu(x)^3$ continuous on the closed interval. The integrated formula therefore gives a continuously differentiable representative of $u'$, and hence $u\in C^2([0,1])$ with
$$
u''(x)=2xu(x)^3.
$$
Differentiate once more. The right side is continuously differentiable because $u\in C^2$, and
$$
\boxed{u^{(3)}(x)=2u(x)^3+6xu(x)^2u'(x).}
$$
Thus $\boxed{u\in C^3([0,1])}$. These formulas give continuous one-sided derivatives at the endpoints as well. Repeated bootstrapping would actually give smoothness of every order.
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