Smooth extension from bounded endpoint jets

Statement

Under the hypotheses of one-sided endpoint jet limits, a smooth $f$ on $U\times(0,T)$ admits a smooth extension through time $T$ , locally in $U$ . If all its slices have support in one fixed compact $K\subset U$ , the extension for $t\ge T$ can retain that spatial support and vanish for $t\ge T+1$ .

Match the normal jet

Let $F_j(x)=\lim_{t\uparrow T}\partial_t^jf(x,t)$ . Borel's theorem gives a smooth $g(x,\sigma)$ with $\partial_\sigma^jg(x,0)=F_j(x)$ . Use $g(x,t-T)$ for $t\ge T$ and the original $f$ for $t<T$ . Every mixed derivative has matching one-sided limits. The coordinate-segment fundamental theorem of calculus proves that the glued function is smooth.

For fixed compact support, all $F_j$ retain that support, and the shrinking-cutoff Borel construction retains it while being supported in $|\sigma|\le1$ . This extends the given function, rather than asserting convergence of its formal endpoint Taylor series.