= Solution
An <absolutely one-homogeneous functional> satisfies
$$
J(cu)=|c|J(u)\quad\text{for every scalar }c,
$$
with $J(0)=0$. The printed strictly positive codomain cannot hold at zero; the standard interpretation is a nonnegative, possibly extended-real functional.
For $p\in\partial J(u)$, apply the <subgradient inequality> at $v=0$:
$$
0=J(0)\geq J(u)-\langle p,u\rangle,
$$
so $\langle p,u\rangle\geq J(u)$. Apply it at $v=2u$ and use one-homogeneity:
$$
2J(u)=J(2u)\geq J(u)+\langle p,u\rangle,
$$
which gives the reverse inequality. Therefore
$$
\boxed{J(u)=\langle p,u\rangle.}
$$
This is the <Euler identity for a convex one-homogeneous functional>; differentiability is unnecessary.
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