= Discrete antidifferentiation on the p-adic integers
{title2=$Jf(x)=\sum_{n\geq0}a_n(f)\binom{x}{n+1}$}
The <Mahler coefficients> tend to zero, so the displayed series defines a <continuous function> and has value zero at zero. The <Pascal's identity> and uniform convergence give $\Delta Jf=f$. Thus the <forward difference operator> is surjective on <continuous functions on the p-adic integers>. Its kernel consists of constants: period one implies agreement on the dense nonnegative integers, and <continuity> then implies constancy. This selects the unique <discrete antiderivative> vanishing at zero.
There is also a direct construction on <locally constant functions>. If $h$ has period $M=p^r$, put $a_j=h(j)$, $S=\sum_{i=0}^{M-1}a_i$, and $P_j=\sum_{i<j}a_i$. For $x=j+My$ with $0\le j<M$ and $y\in\mathbb Z_p$, set
$$
Jh(x)=yS+P_j.
$$
This is a <continuous function> on each residue class. Increasing $j$ gives $Jh(x+1)-Jh(x)=a_j$; at $j=M-1$, the next point has residue zero and quotient $y+1$, giving the same identity. The <ultrametric inequality> gives $\|Jh\|_\infty\le\|h\|_\infty$, and $Jh(0)=0$. The construction is independent of the chosen period, since two normalized <discrete antiderivatives> agree on the nonnegative integers and then on the <p-adic integers> by <continuity>. It is linear on the <locally constant functions>, which form a <dense subset> in the <supremum norm>. Completeness therefore extends it to every <continuous function>, retaining $\Delta J=I$ and the norm bound. This proves surjectivity without first using the <Mahler theorem>.
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