This is true for inverse images but not for images. Consider the example of a parabola, which certainly represents a continuous function:

f(x) =

Then the image of the set (-1, 1) is the set [0, 1). That set is neither open nor closed; in particular, it is not open.

To find a counterexample for images of closed sets, let's look at the following function:

This function is continuous on the whole real line, and the image of the set [0, ) is the set (0, 1]. Therefore we have found a closed set whose image under a continuous function is not closed (nor open).