Energy of a piecewise smooth curve

By [F1], $\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_g^2$ is continuous and nonnegative on every smooth piece, so every displayed Riemann integral is finite and nonnegative. If two subdivisions are used, their union is a finite common refinement. Ordinary finite additivity of the Riemann integral splits the integral of $\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_g^2$ over each old piece into the integrals over its refined subintervals, so both sums equal the common-refinement sum. Changing one-sided derivative conventions at finitely many corners does not change any integral. Thus $E(\gamma )$ is well-defined and nonnegative.

A constant curve has speed and energy zero. For a singleton parameter interval the empty sum is zero; no negative-length interval is admitted. In dimensions zero and one the same formula applies, with every zero-dimensional curve locally constant. An empty target admits no nonempty-domain curve. Endpoints contribute only through the integrals and their values at the two individual endpoints do not affect them. Only a given finite subdivision and its finite common refinement are used, so no choice axiom is needed.