Is the image of a closed set under a contiuous function closed?

Question: Is the image of a closed set under a continuous function always closed?

Answer: No.

Why: Consider the projection of the graph of $y = \frac{1}{x}$ onto the $x$-axis, or simply the inclusion of an open interval into ${\bf R}$ (it is closed in itself!)...

Also. is the image of an open set under a continuous function open?