Introduction to Mathematical Analysis I - Second Edition

99 Exercises 3.7.1 Let f be the function given by f ( x ) = ( x 2 , if x 6 = 0; − 1 , if x = 0 . Prove that f is lower semicontinuous. 3.7.2 Let f be the function given by f ( x ) = ( x 2 , if x 6 = 0; 1 , if x = 0 . Prove that f is upper semicontinuous. 3.7.3 Let f , g : D → R be lower semicontinuous functions and let k > 0 be a constant. Prove that f + g and k f are lower semicontinous functions on D . 3.7.4 I Let f : R → R be a lower semicontinuous function such that lim x → ∞ f ( x ) = lim x →− ∞ f ( x ) = ∞ . Prove that f has an absolute minimum at some x 0 ∈ R .