Abstract
Any continuous time function on aC k space-timeV (i.e., a scalar onV that increases along any causal curve) can be approximated by smoothC k time functions. A time function defined on a (bounded) subset of a stably causalV can be extended to a time function on the whole ofV.
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Seifert, H.J. Smoothing and extending cosmic time functions. Gen Relat Gravit 8, 815–831 (1977). https://doi.org/10.1007/BF00759586
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DOI: https://doi.org/10.1007/BF00759586