Paper 1, Section I, E

Analysis I | Part IA, 2012

What does it mean to say that a function f:RRf: \mathbb{R} \rightarrow \mathbb{R} is continuous at x0Rx_{0} \in \mathbb{R} ?

Give an example of a continuous function f:(0,1]Rf:(0,1] \rightarrow \mathbb{R} which is bounded but attains neither its upper bound nor its lower bound.

The function f:RRf: \mathbb{R} \rightarrow \mathbb{R} is continuous and non-negative, and satisfies f(x)0f(x) \rightarrow 0 as xx \rightarrow \infty and f(x)0f(x) \rightarrow 0 as xx \rightarrow-\infty. Show that ff is bounded above and attains its upper bound.

[Standard results about continuous functions on closed bounded intervals may be used without proof if clearly stated.]

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