A $C^{k+r}$ residual vanishing through order $r-1$ has a $C^k$ scaled extension

For each parameter multi-index $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }\le k$, iteratively apply the chain rule [F2] to the identities ${\mathrm{\partial }}_h^{j}f(0,\alpha )=0$. Thus ${\mathrm{\partial }}_{\alpha }^{\beta }{\mathrm{\partial }}_h^{j}f(0,\alpha )=0$ for every $0\le j<r$, as needed below.

For each parameter multi-index $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }\le k$, put $k_{\beta }=k-\mathrm{\mid }\beta \mathrm{\mid }$, $N=r+k_{\beta }$, and $g_{\beta }(h)={\mathrm{\partial }}_{\alpha }^{\beta }f(h,\alpha )$. For every $s\le k_{\beta }$, put $m=N-s$. If $m\ge 2$, apply [F1] componentwise to $g_{\beta }^{(s)}$ through degree $m-1$; continuity of its highest derivative lets us replace the Lagrange coefficient at $\theta h$ by its value at zero plus $o(1)$. If $m=1$, subtract the linear Taylor polynomial: the derivative of the remainder is $o(1)$ by continuity, so [F3] gives a remainder $o(\mathrm{\mid }h\mathrm{\mid })$. Thus in every case $g_{\beta }^{(s)}(h)$ equals its Taylor polynomial through degree $m$ plus $o(\mathrm{\mid }h{\mathrm{\mid }}^m)$. By step 1.1 all coefficients of degrees below $r$ vanish. For $h\mathrm{\ne }0$, the quotient rule expresses the $\mathrm{\ell }$th derivative of $g_{\beta }\mathrm{/}{h}^r$ as a finite linear combination of $h^{-r-\mathrm{\ell }+s}{g}_{\beta }^{(s)}(h)$, $0\le s\le \mathrm{\ell }$. Substituting the expansions shows that the remainder terms are $o(\mathrm{\mid }h{\mathrm{\mid }}^{k_{\beta }-\mathrm{\ell }})$; the polynomial part has a continuous extension whose $\mathrm{\ell }$th derivative at zero is $\mathrm{\ell }!{\mathrm{\partial }}_h^{r+\mathrm{\ell }}{\mathrm{\partial }}_{\alpha }^{\beta }f(0,\alpha )\mathrm{/}(r+\mathrm{\ell })!$. Continuity of the derivatives on compact parameter neighborhoods makes these limits locally uniform in $\alpha$.

Define the value at $h=0$ by the displayed limit formula. For each $\mathrm{\mid }\beta \mathrm{\mid }+\mathrm{\ell }\le k$, step 2.1 gives a locally uniform limit for the corresponding derivative as $h\to 0$; at $h=0$ these formulas are differentiable in $\alpha$ because $f$ is $C^{k+r}$, and their $h$-derivatives agree with the next displayed coefficient. Induction on total derivative order therefore shows that all mixed partial derivatives through order $k$ exist and are continuous across $h=0$. The extension is $C^k$ with the asserted formula.