Exercise 1.5.4

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 an open set $V \subseteq \mathbb{R}$ such that the image, $f(V)$, of $V$ is not open.

Prakash Anand · Accepted Answer

Take $f(x)=0 \forall x$. Clearly $f$ is continuous, and if we look at the at open set $(0,1)$ it gets mapped into $\{0\}$ which is clearly not open (with respect to $(\mathbb{R}, d)$ ).

Exercise 1.5.4

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Exercise 1.5.4

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