Let f be a real valued function. For f, we define
#ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20f%5e*%28x%29=%5cinf%5cleft%5c%7b%5csup_%7by%5cin%20N%7df%28y%29:N%5ctext%7b%20is%20a%20nbd.%20of%20%7dx%20%5cright%5c%7d%5c%5d%7d%25.png);
#ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20f_*%28x%29=%5csup%5cleft%5c%7b%5cinf_%7by%5cin%20N%7df%28y%29:N%5ctext%7b%20is%20a%20nbd.%20of%20%7dx%20%5cright%5c%7d%5c%5d%7d%25.png);
f is normal lower sem?continuous (n.l.s.c.) if &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$f%5e*$%7d%25.png); is real-valued and &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$f=%28f%5e*%29_*$%7d%25.png);.