Let f, g be functions defined on [0, 1], with their ranges contained in [1,1]. (a)...

Transcribed Image Text:

Let f, g be functions defined on [0, 1], with their ranges contained in [1,1]. (a) Prove that sup{f(x) +g(x): ze [0, 1]}<sup{f(a): re [0, 1]}+sup{g(x): xe [0, 1]}. (10) (b) Is it true for all f, g that sup{f(x)+g(1) : r € [0, 1]}= sup{f(x): x € [0, 1]}+sup{g(x) : x= [0, 1]}? (11) If yes, prove it; otherwise, give a counterexample. Think about why it is different from the last question. Let f, g be functions defined on [0, 1], with their ranges contained in [1,1]. (a) Prove that sup{f(x) +g(x): ze [0, 1]}<sup{f(a): re [0, 1]}+sup{g(x): xe [0, 1]}. (10) (b) Is it true for all f, g that sup{f(x)+g(1) : r € [0, 1]}= sup{f(x): x € [0, 1]}+sup{g(x) : x= [0, 1]}? (11) If yes, prove it; otherwise, give a counterexample. Think about why it is different from the last question.