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(a) Suppose f: (a, b) → R is continuous. Show that if ƒ(a+) and ƒ(b¯) are finite(i.e. < ∞), then ƒ is bounded on (a, b).
Posted: Sat Jul 09, 2022 2:25 pm
by answerhappygod
- A Suppose F A B R Is Continuous Show That If F A And F B Are Finite I E Then F Is Bounded On A B 1 (97.72 KiB) Viewed 80 times
(a) Suppose f: (a, b) → R is continuous. Show that if ƒ(a+) and ƒ(b¯) are finite(i.e. < ∞), then ƒ is bounded on (a, b). Argue whether the converse is true. [C5, 5 marks] (b) Assume the following
statement: If a function f : (a,b) → R is uniformly continuous, then both f(a+) and ƒ(b¯) are finite. (This is true. Take as granted. Proof omitted.) Can we change the term uniformly continuous to continuous in the assumption? Justify - prove if yes, give a counterexample if no. [C5, 2 marks]