Algebra: A circuitous path

Imagine taking a walk in a city by a river. The river passes through the city and branches around some islands, and several bridges connect the opposite banks to the islands and each other. The image below should help you to picture this.





The challenge: Is it possible to walk about the entire city, crossing each bridge once and only once? You do not need to start and end your walk at the same place.

For the prize, you'll need either an example path or else a solid explanation of why there is no such path.

Once you have your answer, look for the pattern to explain why you can or cannot make the journey. If you can, what would happen if one of the bridges were closed? If you can't, what place(s) would a bridge be necessary in order to make it possible?