Paper 1, Section I, E

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

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

The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and non-negative, and satisfies $f(x) \rightarrow 0$ as $x \rightarrow \infty$ and $f(x) \rightarrow 0$ as $x \rightarrow-\infty$. Show that $f$ 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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