1. A continuous function and a set whose image is not connected.
2. A continuous function and a disconnected set whose image is connected.
3. A function such that the image of a connected set is disconnected.
4. Is it true that inverse images of connected sets under continuous functions are again connected ?

1. If f is a continuous function, the image of every connected set is connected. Therefore, to find an example for this situation, we must start with a disconnected set. It is very easy then to come up with examples. One can use, for example, the standard parabola and two separate intervals on the positive real axis. Details are left as an (easy) exercise.

2. Again, we can use the standard parabola. By taking two suitable intervals, one on the positive real axis, and the other on the negative real axis, one can easily construct such an example. Details are left as an exercise.

3. This time we can not take a continuous function, since the images of connected sets always will be connected for those types of functions. An easy example may be provided by taking the function f(x) = 1 for x > 0 and f(x) = -1 for x 0 and a suitable connected interval on the real axis. Details, as usual, are left as an exercise.

4. No, that is not true: inverse images of connected sets may be disconnected. A quick look at the standard parabola will provide us with an easy example. Details ? Of course left as exercise.