In Reply to: Continuous functions on disconnected sets? posted by David Molnar on November 27, 1999 at 15:18:34:
Glad to hear other opinions and examples concerning this question. When in doubt, always go back to the definitions. Is your example
continuous at each point in its domain? Did any of our approaches to continuity mention domain restrictions?
The Connected Image Theorem does say that the the image of a connected set under a continous map must be connected. If, as your second example suggests, some given function has an image that is not connected, does it or does it not necessarily follow that the function is discontinuous?
Imagine taking the real line with the discrete topology rather than the standard Euclidean one. Is it connected? What about continuous functions on this space?