- what is the image of [-2, 1] ?
- find a closed set whose image is not closed
To find a closed set whose image is not closed we must consider an unbounded set. Otherwise, a closed and bounded set is compact, and since the image of a compact set under continuous functions is compact, it is in particular closed again.
But if we look at the closed, unbounded set [0, ), we see that the image of that set under the above function is the set (0, 1], which is not closed.