Exercise 1.5.5

Question

Let $(\mathbb{R}, d)$ be the real line with the standard metric. Give an example of a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, and a closed set $F \subseteq \mathbb{R}$ such that $f(F)$ is not closed.

Prakash Anand · Accepted Answer

Take $f(x)=e^{x} \forall x . f$ is continuous, and if we take the closed set $(-\infty, 0]$ it gets mapped into $(0,1]$ which is not closed.

Exercise 1.5.5

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Exercise 1.5.5

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