I have the following proposition from A course in $p$-adic analysis by A. Robert, Proposition 2 (page 220) under differentiation.
Can someone show me how (ii) implies (i) and (ii) implies (iii) in the proof?
where $\phi f$ and $S^1$ is denoted as follows:
If $\widetilde{\Phi}$ as in (ii) exists, then for each $a\in X$, by continuity $\Phi(a,a)$ must be equal to the limit $$\lim_{(x,y)\to(a,a)}\widetilde{\Phi}(x,y).$$ In particular, restricting this limit to $(x,y)\in X\times X-\Delta_X$ so $\widetilde{\Phi}(x,y)=\Phi f(x,y)$, the existence of this limit says exactly that $f\in S^1(a)$ (and the limit is by definition $f'(a)$, so $\Phi(a,a)=f'(a)$).
Moreover, defining $\alpha(x,y)=\widetilde{\Phi}(x,y)-f'(x)$, the given proof of Proposition 2 shows that $f(y)=f(x)+(y-x)f'(x)+(y-x)\alpha(x,y)$ is true for $x\neq y$, and it is also true when $x=y$ trivially. Moreover $\alpha$ vanishes on $\Delta_X$ since $\Phi(x,x)=f'(x)$, and $\alpha$ is continuous since $\Phi$ is (and $f'$ is continuous since $f'(x)=\Phi(x,x)$), so this proves (iii).