Constructing a Continuous Function with 2 Different Range Values

[muzihc]

Homework Statement

Provide an example of f:D-->R which is continuous but whose range has two different numbers only.

Homework Equations

The Attempt at a Solution

For the range to have only two different values, it's seems impossible to construct a continuous function without jumps. In fact, it seems like the intermediate value theorem could prove this.

I've asked my professor if this is possible, he says it is, and that studying the definitions of continuous closely will reveal it.

Then I've asked a few friends, who are very good at math, and they say it isn't. I also asked Dr. Math, and got a response from Dr. Tom, who says it isn't.

 

[snipez90]

In the usual topology, any function from a discrete set is continuous.

 

[muzihc]

In the usual topology, any function from a discrete set is continuous.

Thanks

I feel like a moron.