Characterize $|\nabla f|$ as minimal function which satisfies an upper gradient inequality

Question

Let $f \in C^1( \mathbb R^n, \mathbb R) .$ Then one by chain rule has $$ (*)\qquad |f(g(1))-f(g(0))| \leq \int_0^1 |\nabla f|(g_t)|g'(t) |\ dt, \quad \forall g \in C^1([0,1],\mathbb R^n). $$ I have heard in a lecture that $|\nabla f|$ can actually be characterized as the smallest continuous function for which $(*)$ holds. Is this true? How can I show it?