Is the generalized directional derivative additive?

Question

Let $X$ be a $\mathbb R$-Banach space and $f,g:X\to\mathbb R$ be locally Lipschitz continuous on a neighborhood of $x\in X$. The generalized directional derivative of $f$ at $x$ in the direction $v\in X$ is defined as $$f^\circ(x;v):=\limsup_{\substack{y\to x\\t\to0+}}\frac{f(y+tv)-f(y)}t.$$

Are we able to show that $(f+g)^\circ(x;v)=f^\circ(x;v)+g^\circ(x;v)$?

By the usual subadditivity of $\limsup$ we've got at least "$\le$".

Answer 1

Let $f = -g = x\sin(\frac{1}{x})$ which is Lipschitz. Then, both $f^{\circ} (0, 1) = g^{\circ} (0, 1) = 1$, whereas $(f+g)^{\circ} (0, 1) = 0$