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We study for which polynomials $F$ a thin shell wormhole with a continuousmetric (connecting two Schwarzschild spacetimes of the same mass) satisfy thenull energy condition (NEC) in $F(R)$-gravity. We avoid junction conditions byusing the mathematical framework of the Colombeau algebra which describes ageneralized framework of the distributional geometry such that one can definemultiplications between distributions generalizing the tensor product of smoothtensors. The aim for physics is to motivate a conjecture about the satisfactionof the NEC for suitable quadratic $F$ while the aim for mathematics is toderive a rigorous framework describing this situation. Here the $F(R)$-gravityshould be seen as a toy model, important is that the NEC may be satisfied bysome form of "microstructure" which does not arise in the classical setting andmay have interesting physical meanings.

Existence of Thin Shell Wormholes using non-linear distributional geometry