Well to me it was defined as such (terminology might vary from my language to english) :
An algebra over a field $\Bbb K$ is a quadruplet $(A, +, \times, \cdot)$ where :
$(A, +, \times)$ is a ring
$(A, +, \cdot)$ is a vector space over $\Bbb K$
$$\forall (u, v)\in A^2, \forall \lambda \in \Bbb K, \lambda \cdot (u\times v) = (\lambda\cdot u)\times v = u\times(\lambda\cdot v)$$
How can we get from the system
\begin{align*}
\lambda_t + u\lambda_x &= -u_x,\\
u_t + uu_x &= -\varphi'(\rho)\lambda_x,
\end{align*}
where $\lambda = \log \rho$, to
$$f(\rho) = \int\varphi'(\rho)\mathop{\mathrm{d}\lambda}?$$
I've tried replacing the $u_t$ and $u$ in the first equation and doing...
Consider the fabius function
https://en.m.wikipedia.org/wiki/Fabius_function
https://people.math.osu.edu/edgar.2/selfdiff/
How does one show that this function is nowhere analytic ?
Probably related , Maybe even a step in the answer : how to evaluate this function for nonreals ? Is it defined...
