A Parametric Optimization Point-Of-View of Comparison Functions

The paper’s Theorem 2 asserts (i) if f and g are lower semicontinuous then f∨g is lower semicontinuous, and (iv) if f is continuous and g is lower-semicontinuous and level-bounded then f∨g is continuous, both stated without domain qualification. These are false on R: with g(x)=x^2 (lsc) and f(x)=0 at x=0 and 1 elsewhere (lsc), F(s)=sup_{g≤s}f has F(s)=-∞ for s<0, F(0)=0, and F(s)=1 for s>0, so F is not lsc (nor continuous) at s=0. The proof outline in the paper relies on an incorrect use of set-valued lower hemicontinuity, effectively replacing [g≤s] by ⋃_{n}[g≤s_n] as s_n↗s (which yields [g<s], not [g≤s]); see their use of Berge’s theorem for item 1 and continuity claim, and the argument about M(s)=⋃M(s_n) in the proof sketch . By contrast, the candidate solution correctly flags (i) and (iv) as false as stated, adds the necessary domain restriction for (ii) (“finite-valued on [inf g,∞), max attained”) which the paper’s proof implicitly uses anyway, and gives a simpler correct proof of (iii) (divergence to +∞ when f is level-bounded). For part B, the candidate’s sequential compactness proof matches the paper’s Theorem 3 (extended lsc for f∧g) assumptions and conclusion .