IN THIS PAPER THE CONCEPT OF DUAL COSMOLOGICAL TIME FUNCTION AND ITS REGULARITY IS INTRODUCED. IT IS SHOWN THAT THE REGULARITY OF COSMOLOGICAL AND DUAL COSMOLOGICAL TIME FUNCTIONS ARE INDEPENDENT OF EACH OTHER. IT IS PROVED THAT IF THE COSMOLOGICAL TIME FUNCTION OF THE SPACETIME (M, G) IS CONTINUOUS, T®0 ALONG EVERY PAST INEXTENDIBLE CAUSAL CURVE AND T®¥ ALONG EVERY FUTURE INEXTENDIBLE CAUSAL CURVE THEN (M, G) IS GLOBALLY HYPERBOLIC.