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The condition says that, along a piecewise smooth extremal x(t) (i.e. an extremal which is smooth except at a finite number of corners) for an integral , the partial derivative must be continuous at a corner T. That is, if one takes the limit of partials on both sides of the corner as one approaches the corner T, the result must be the same answer.
The condition allows one to prove that a corner exists along a given extremal. As a result, there are many applications to differential geometry. In calculations of the Weierstrass E-Function, it is often helpful to find where corners exist along the curves. Similarly, the condition allows for one to find a minimizing curve for a given integral.