Lectures On the Calculus of Variations

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Is a continuous closed curve without mul- tiple points (a so-called ''^Jordan-curve''') . It divides, there- fore, ^ the oc, ?/-plane into an interior and an exterior. According to a theorem due to Schoenfliess" the set of points §>j. Is identical ivitli the interior of 2' together witli the houndarji 2'.
2. Let /q, iy be two values of / satisfying the inequality and let @o denote the arc of the curve (10) corresponding to the interval (/o, ti). Since the line: 0.^:0^, to^t^t^ lies in the inter
...ior of iSj^. , its image @o li^s in the interior of ^^. And has, therefore, no point in common with the boundary 2'. The two curves ®o ^^^^ ^' being continuous, it follows, '" therefore, that a neighborhood (/o) of the arc @o c^in he con- structed which is entirely contained in ^f..
3. Since A(/, Oo)=t=0 in {TqTi) and A(/, a) is continuous in iSfc, it follows from the theorem on uniform continuity* that k can he taken so smcdl that 1 Compare J. I, No. 102. The interior as well as the exterior is a " continuum.


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