 Fourier decay from a derivative jump

For a piecewise smooth decaying function, a jump in its $j$th derivative at $a$ contributes the displayed algebraic term to its high-frequency Fourier transform. Distributionally, $D^{j+1}f$ contains $[f^{(j)}{]}_a\delta (x-a)$, and integration by parts or the derivative-transform identity supplies the factor $(ik{)}^{-j-1}$. Adequate integrability of the subsequent derivatives controls the remainder. Smooth localized polynomial terms give no algebraic cusp contribution.