Funtion continuity and open sets

[complexnumber]

Homework Statement

Suppose that $$f : (X,d_X) \to (Y,d_Y)$$. If $$f$$ is continuous,
must it map open sets to open sets? If $$f$$ does map open sets to
open sets must $$f$$ be continuous?

Homework Equations

The Attempt at a Solution

The answer to the first question is yes. The answer to the second question I guess is "no". Is this correct? How can I prove it?

 

[HallsofIvy]

Neither is true. For example, a constant function is trivially continuous but maps any set, including any open set, to a singleton set which is not, generally, open.

Conversely, if B has the discrete topology, so that all sets are open, and A is a [0, 1] with the usual topology, the function f(x)= a for all $0\le x< 1/2$ and f(x)= b for all $1/2\le 0\le 1$, where a and b are distinct points in B, is not continuous but maps open sets to open sets.