Show continuous functions need not be open maps and open maps need not be continuous.

Question

A function from one metric space to another is said to be an open map if it maps open sets to open sets. Similarly one can define a closed map.

1-Provide a continuous function which does not map an open set to another open set?

2- Provide a function which maps every open set to another one but it is not a continuous function?

Answer 1

For 1) take a constant function and let it be that singletons in the codomain are not open.

For 2) take the identity $\mathbb R\to\mathbb R$ where codomain is discrete (so $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ otherwise) and domain is not discrete.

Answer 2

A) To violate openness you can give any map which is not bijective.

B) Open maps which are not continuous