Let

f, 9 be real functions, and for each x E Dom

(f) n Dom (g) define

(f V g)(x) := max{f(x), g(x)} and (f 1\ g)(x) := min{f(x), g(x)}.

(

a) Prove that and

(f V g)(x) = (f + g)(x) + l(f - g)(x)1 2

(f 1\ g)(x) = (f + g)(x) -I(f - g)(x)1 2

for all x E Dom

(f) n Dom (g).

(b) Prove that if L = lim f(x) and M = lim g(x)

x~a x~a exist, then (f V g)(x) ---; Lv M and (f 1\ g)(x) ---; L 1\ M as x ---; a.