Continutiy and intervals?

Question

I need help with the following proof:

If the domain of a continuous function is an interval, show that the image is an interval.

This doesn't seem like it should be that hard, but I'm having trouble putting it into "math" terms. Thanks.

Answer 1

This follows from the general result that continuous functions preserve connectedness. On the real line, the connected sets are just the intervals, including the so called degenerated intervals, that is, those of the form [a].

If f:I --> R is continuous on the interval I, then, since I is connected, so is f(I). Therefore f(I) is an interval.