Let g R times 0 infinity rightarrow R be defined by gx1x2 x

Let g: R times (0, infinity) rightarrow R be defined by g(x^1,x^2) = |x^1,x^2 - 1|, f: R rightarrow R by f(x^1):. Show that f is not lower semicontinuous.

Solution

f(0) = inf_{x>0} g(0,x) = inf |0-1| = 1

for z > 0

f(z) = inf_{x>0} g(z,x) = inf_{x>0} |zx -1| = 0

for z<0

f(z) = 1

So f is not lower semicontninuous at 0