From page 204 of this book: a function $f\colon X \to Y$ between Banach spaces is called Newton differentiable at $x$ if $$\lim_{h \to 0}\frac{\lVert f(x+h)-f(x)-f'(x+h)(h)\rVert}{\lVert h\rVert} =0$$ and $f'(x) \in \mathcal{L}(X,Y)$.
In page 213, Example 14.13, the author says that $f\colon \mathbb{R} \to \mathbb{R}$ defined $f(x) = \max(0,x)$ has Newton derivative $$f'(x)(h) = \begin{cases} 0 &: x < 0\\ \delta h &: x = 0\\ h &: x >0 \end{cases}$$ where $\delta \in [0,1]$. Does something go wrong if we take $\delta$ outside of $[0,1]$? It looks like it doesn't matter what the derivative does at $x=0$ and the definition should hold. Is there a good reason the author chooses $\delta$ in that interval?